# Experimental Analysis on Dynamic Response ofX-Section Piled Raft Composite Foundation under Cyclic Axial Load for Ballastless Track in Soft Soil

Authors: Qiang Fu
Journal: Shock and Vibration
 (2021)

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## Abstract

Large-scale model tests were established at a scale of 1/5 using a 7 m deep model tank with cross-sectional dimension of 5 m × 4 m, to study the vibration response characteristics of ballastless track, embankment, andX-section piled raft foundation under cyclic axial load, including the vibration displacement, velocity, dynamic soil, and pile stress. Cyclic dynamic loading can be achieved by controlling the loading frequency and cycles through the vibration servo control loading system. The test results are presented in the variation of dynamic displacement, velocity, and stress of X-section piled raft composite foundation. The vibration displacement, velocity, and stress of the track, embankment, and pile foundation follow a pattern of vibration characteristics of loading sine wave. The vibration characteristics of loading waves can be identified easily from the peaks and troughs in the dynamic response of displacement, velocity, and stress at many locations of track slab, embankment, cushion, and underlying soil, at which the vibration response presents almost monotonically increasing tendency with the loading frequencies. With the increase of loading frequency, the vibration responses at the track structure and embankment have higher increasing rates than those at substructure (raft, cushion, and subsoil). The piled raft bears more dynamic load than cushion and subsoils, to ensure long-term dynamic stability and safety of the foundation soils. The model testing results provide a better understanding of the dynamic response characteristics of ballastless track, embankment, and X-section piled raft foundation under cyclic axial load in soft soil.


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## Body

## 1. Introduction

In recent years, with the rapid development of high-speed railway, the studies on the dynamic response of ground and environmental vibration induced by moving trains have attracted many attentions. Ballastless slab track was proven to be a substantial solution to the high stability, high reliability, high maintainability, and high serviceability requirements of high-speed railways as compared with the traditional ballasted track [1–3]. In ballastless tracks, concrete slabs are adopted to replace the ballast layer used in the traditional ballasted tracks, which can reduce the maintenance costs by 70–90% and reduce the weight and height of the track structure. Since the ballastless slab track has greater lateral and longitudinal resistance, it can effectively maintain the track geometry, and the possibility of track bending is reduced [4, 5]. In China, ballastless tracks have been used extensively in newly built areas of China as effective alternatives to traditional ballasted railway tracks because of their high durability and low whole-life cost. The operating speed of high-speed trains produces large vibration loading on the tracks, embankment, and soil foundations. The train running comfort, safety, and durability depend on the dynamic characteristics and bear capacity of track, embankment, and underlying soil foundations. Varieties of research about the dynamic response of track, embankment, and soil foundations subjected to dynamic loads have been conducted through experimental solutions [6–14], numerical solutions [15–20], and analytical solutions [21–24]. Generally, a considerable amount of researches have been carried out on the dynamic response of surrounding structures and environment to train loads. Most studies concern the dynamic response of the track structure and embankment, as Bian et al. summarized [12]. Moreover, in spite of the increasing use of piled foundations as a rapid construction technique for highway and railway lines, most studies focused on investigating the bearing capacity and behavior of foundations without improvement under dynamic loads. In addition, very limited attention has been paid to the bearing capacity and behavior of these piled foundation systems under dynamic loads.With the development of social economy and the high rate of population growth, high-speed vehicles and systems are proposed to address these issues; increasing construction activities are presented in these unsuitable areas, such as coastal, low-lying marsh, and other soft soil areas. Soft soils are characterized by high water content, high compressibility, low bearing capacity, large deformation, and low shear strength, etc. The sustainable performance of transportation infrastructure on these soils requires innovative construction and ground improvement techniques. Embankments are widely used to elevate the ground level for the construction of railways on soft soils. Piled raft foundations (PRFs) have received considerable attention in recent years. In soft soil areas, the piled raft composite foundation, as a ground treatment method, is more cost-effective and has less construction time than others and has been widely used to support embankments over soft soil around the world. The piled raft structure forms an optimized and efficient foundation type, which can improve the bearing capacity and reduce the total and differential displacement, wherein the load carrying capacities of both raft (pile cap) and piles are utilized [25]. The studies of working performance, bearing capacity, load transfer mechanism, and related influencing factors for piled raft structures have been reported [26–29]. Dynamic raft-pile group interactions related to load characteristics and bearing capacity are also considered and analyzed [30, 31].Recently, foundation improvement by usingX-section cast-in-place concrete (XCC) piles has been developed and applied in highway construction [32–34], which has been applied to improve the soft soil foundation of the Jing-Hu railway in Nanjing of China. Liu et al. [32] established a simplified analytical model, which can be used to interpret and predict the displacement, stress, and excess pore pressure caused by the X-section pile installation in soft soil. Lv et al. [33] investigated the stress transfer and load sharing mechanisms of an X-section cast-in-place concrete (XCC) piled raft, using numerical and field tests. However, the above studies focused on the behavior capacity and load transfer mechanism of X-section piled raft foundation under static loads. Sun et al. [34] presented a physical model of a ballastless railway to simulate the dynamic response of BTXPR foundation on air-dried and saturated sand subsoils. Very limited attentions were considered to investigate the dynamic behavior of track, embankment, and X-section piled raft foundation in soft soil under dynamic loads.This paper presents a physical model of ballastless railway track, embankment, andX-section piled raft foundation in soft soils. The short-term dynamic response of each position of the system is tested and recorded to study the vibration response characteristics and laws of the X-section piled raft composite foundation under cyclic axial load, including the vibration response, transmission and attenuation of vibration displacement, velocity, dynamic soil, and pile stress.


## 2. Large-Scale Model Test

### 2.1. Model Design

As is shown in Figure1, the large-scale model of XCC piled raft foundation for ballastless railway was established in a big model tank of 5 m length, 4 m width, and 7 m height. The X-section piles were constructed using C25 concrete. Figure 1 shows the cross section of a constructed X-section pile. The three cross-sectional parameters, namely, a, R, and θ, are denoted as the open arc spacing, the cross-sectional radius, and the open arc, respectively. As is shown in Figure 1, the geometric parameters of the X-section pile are 2a, 2R, and θ, where 2a is the length of the flat sides, 2R is the X-section pile, θ is the angle between tangent drawn from the center of two adjacent sides. The geometric parameters 2a, 2R, and θ are 39 mm, 157 mm, and 90°, respectively.Figure 1
Cross section of a constructedX-section pile.Layout of the instruments for model test in the dynamic loading system is shown in Figure2. The concrete base was a layer of 0.06 m thickness to support the track slab. The roadbed under the concrete base was a layer of 0.08 m to support the concrete base. The embankment was a layer 0.46 m thick AB granular below the roadbed. The concrete raft was 0.12 m thick to support the railway track system. The gravel cushion was 0.06 m thick below the raft. The length of the X-section pile was 4.3 m below the gravel cushion. The piles were arranged with an equal spacing of 0.6 m. The underlying silty soil was 4.3 m thick along with the piles. The supporting layer was 1 m thick and composed of sand with a certain particle size distribution. Velocity sensors were placed at various locations along the horizontal (V1–V5) and vertical directions (V6–V10) of the X-section piled raft foundation with a constant distance. Displacement sensors were placed at the top of track slab D1 and roadbed D2. The earth pressure sensors (E1–E7) were placed t in the middle cross section of the piled raft foundation at various locations with a constant distance. Strain sensors were placed at various locations along the center (S1–S3) and side pile (S4–S6), respectively. The rigid boundary conditions will affect the transmission and attenuation of vibration waves along the soil and cause certain blocking and reflection between model tank and soil. The rigid boundary conditions may have influence on test results. Therefore, geotechnical damping materials such as asphalt and geomembrane are used between model tank and soil to reduce the influence of rigid boundary on model test results.Figure 2
Layout of instruments for model test in dynamic loading system. (a) Cross-sectional view. (b) Plain view.
(a)(b)


### 2.2. Materials Properties

The particle size distributions of these geomaterials (gravel, granular soil, sand, silty soil, and supporting layer) are plotted in Figure3. The physical properties and parameters of the geomaterials used are summarized in Table 1. The soft soils around the X-section piles are silty soil with low strength, low compression modulus, and poor dynamic response characteristics. The above particle size distributions are obtained by model test and are consistent with the characteristics of actual engineering material in high-speed railway.Figure 3
Particle size distributions of geomaterials in ballastless railways.Table 1
Physical properties and parameters of geomaterials used.
Filling materialsγ (kN/m3)w (%)Dr (%)Liquid limit (%)Plastic limit (%)Plastic indexCuCcSilty soil18.627.8—31.024.16.92.761.71AB granular21.95.3563———62.89Gravel cushion23.517.0792———19.231.34Supporting layer (sand)19.4325.569.5———2.420.93


### 2.3. Applied Cyclic Axial Load

The vibration servo control loading system consisted of a vibration excite and a signal acquisition system. Bian et al. [11] presented a full-scale model test with simulated train moving loads at various speeds up to 360 km/h, indicating the frequency contents of track slab, roadbed, and embankment for the raw velocity histories mainly distributed in the range of 0–30 Hz. In this paper, the dominant frequency is mainly distributed within 0–30 Hz, the maximum thrust and tension of the loading system were both 200 kN, and the frequency of load ranged from 0 to 30 Hz. The signal sampling frequency of the acquisition system ranged from 0 to 128 kHz, and the frequency used in the test was 200 Hz. The cyclic axial load was simulated as follows:(1)Qt=Q0+A0sin2πft,where Q (t) is the cyclic axial load; Q0 is the static load; A0 is the amplitude of dynamic load; f is the loading frequency, which ranges from 1 to 30 Hz; t is the cyclic loading period.Based on the design standards of high-speed railway in China, the designed axle load of high-speed train was set to 200 kN, and the design overload value of 250 kN was used in this loading test. In this model, the geometric scale of 1 : 5 was chosen. Similarity ratios of model parameters (stress, frequency, and load) were calculated by Bockinghamπ theorem and are listed in Table 2. Thus, the max value of Q0 and A0 was 10 kN and 5 kN, respectively.Table 2
Model test scale factors.
ParameterGeometryStressVolumeDensityModulusVelocityLoadTimeFrequencyScale factor1/511/1251111/251/55For example,f = 5 Hz, the dynamic loading wave is shown in Figure 4, and T = 0.2 s is the cyclic period. Cyclic dynamic loading can be achieved by controlling the loading frequency and cycles through the vibration servo control loading system. Different loading conditions are executed to study the dynamic response of X-section piled raft composite foundation in soft soil. There is a preloading load in the loading process of the actuator to ensure that the actuator does not break away from the railway track structure. The preloading load is 10 kN, and the medium value of the loading wave is 10 kN. The parameters of the system displacement, velocity, and soil pressure generated by the initial preloading are cleared, and then the dynamic loading cycle is carried out for 100 times. The short-term dynamic response of each position of the system is tested and recorded to study the vibration response characteristics and laws of the X-section piled raft composite foundation under cyclic axial load, including the vibration displacement, velocity, dynamic soil, and pile stress.Figure 4
Dynamic loading wave (f = 5 Hz).


## 2.1. Model Design

As is shown in Figure1, the large-scale model of XCC piled raft foundation for ballastless railway was established in a big model tank of 5 m length, 4 m width, and 7 m height. The X-section piles were constructed using C25 concrete. Figure 1 shows the cross section of a constructed X-section pile. The three cross-sectional parameters, namely, a, R, and θ, are denoted as the open arc spacing, the cross-sectional radius, and the open arc, respectively. As is shown in Figure 1, the geometric parameters of the X-section pile are 2a, 2R, and θ, where 2a is the length of the flat sides, 2R is the X-section pile, θ is the angle between tangent drawn from the center of two adjacent sides. The geometric parameters 2a, 2R, and θ are 39 mm, 157 mm, and 90°, respectively.Figure 1
Cross section of a constructedX-section pile.Layout of the instruments for model test in the dynamic loading system is shown in Figure2. The concrete base was a layer of 0.06 m thickness to support the track slab. The roadbed under the concrete base was a layer of 0.08 m to support the concrete base. The embankment was a layer 0.46 m thick AB granular below the roadbed. The concrete raft was 0.12 m thick to support the railway track system. The gravel cushion was 0.06 m thick below the raft. The length of the X-section pile was 4.3 m below the gravel cushion. The piles were arranged with an equal spacing of 0.6 m. The underlying silty soil was 4.3 m thick along with the piles. The supporting layer was 1 m thick and composed of sand with a certain particle size distribution. Velocity sensors were placed at various locations along the horizontal (V1–V5) and vertical directions (V6–V10) of the X-section piled raft foundation with a constant distance. Displacement sensors were placed at the top of track slab D1 and roadbed D2. The earth pressure sensors (E1–E7) were placed t in the middle cross section of the piled raft foundation at various locations with a constant distance. Strain sensors were placed at various locations along the center (S1–S3) and side pile (S4–S6), respectively. The rigid boundary conditions will affect the transmission and attenuation of vibration waves along the soil and cause certain blocking and reflection between model tank and soil. The rigid boundary conditions may have influence on test results. Therefore, geotechnical damping materials such as asphalt and geomembrane are used between model tank and soil to reduce the influence of rigid boundary on model test results.Figure 2
Layout of instruments for model test in dynamic loading system. (a) Cross-sectional view. (b) Plain view.
(a)(b)


## 2.2. Materials Properties

The particle size distributions of these geomaterials (gravel, granular soil, sand, silty soil, and supporting layer) are plotted in Figure3. The physical properties and parameters of the geomaterials used are summarized in Table 1. The soft soils around the X-section piles are silty soil with low strength, low compression modulus, and poor dynamic response characteristics. The above particle size distributions are obtained by model test and are consistent with the characteristics of actual engineering material in high-speed railway.Figure 3
Particle size distributions of geomaterials in ballastless railways.Table 1
Physical properties and parameters of geomaterials used.
Filling materialsγ (kN/m3)w (%)Dr (%)Liquid limit (%)Plastic limit (%)Plastic indexCuCcSilty soil18.627.8—31.024.16.92.761.71AB granular21.95.3563———62.89Gravel cushion23.517.0792———19.231.34Supporting layer (sand)19.4325.569.5———2.420.93


## 2.3. Applied Cyclic Axial Load

The vibration servo control loading system consisted of a vibration excite and a signal acquisition system. Bian et al. [11] presented a full-scale model test with simulated train moving loads at various speeds up to 360 km/h, indicating the frequency contents of track slab, roadbed, and embankment for the raw velocity histories mainly distributed in the range of 0–30 Hz. In this paper, the dominant frequency is mainly distributed within 0–30 Hz, the maximum thrust and tension of the loading system were both 200 kN, and the frequency of load ranged from 0 to 30 Hz. The signal sampling frequency of the acquisition system ranged from 0 to 128 kHz, and the frequency used in the test was 200 Hz. The cyclic axial load was simulated as follows:(1)Qt=Q0+A0sin2πft,where Q (t) is the cyclic axial load; Q0 is the static load; A0 is the amplitude of dynamic load; f is the loading frequency, which ranges from 1 to 30 Hz; t is the cyclic loading period.Based on the design standards of high-speed railway in China, the designed axle load of high-speed train was set to 200 kN, and the design overload value of 250 kN was used in this loading test. In this model, the geometric scale of 1 : 5 was chosen. Similarity ratios of model parameters (stress, frequency, and load) were calculated by Bockinghamπ theorem and are listed in Table 2. Thus, the max value of Q0 and A0 was 10 kN and 5 kN, respectively.Table 2
Model test scale factors.
ParameterGeometryStressVolumeDensityModulusVelocityLoadTimeFrequencyScale factor1/511/1251111/251/55For example,f = 5 Hz, the dynamic loading wave is shown in Figure 4, and T = 0.2 s is the cyclic period. Cyclic dynamic loading can be achieved by controlling the loading frequency and cycles through the vibration servo control loading system. Different loading conditions are executed to study the dynamic response of X-section piled raft composite foundation in soft soil. There is a preloading load in the loading process of the actuator to ensure that the actuator does not break away from the railway track structure. The preloading load is 10 kN, and the medium value of the loading wave is 10 kN. The parameters of the system displacement, velocity, and soil pressure generated by the initial preloading are cleared, and then the dynamic loading cycle is carried out for 100 times. The short-term dynamic response of each position of the system is tested and recorded to study the vibration response characteristics and laws of the X-section piled raft composite foundation under cyclic axial load, including the vibration displacement, velocity, dynamic soil, and pile stress.Figure 4
Dynamic loading wave (f = 5 Hz).


## 3. Test Results and Discussion

### 3.1. Dynamic Displacement Response

Cyclic axial loading on the track slab resulted in the dynamic displacement of the track slab and underlying structures. Time histories and frequency contents of the vibration displacement of track slab (D1) and roadbed (D2) measured in the physical model are plotted and shown in Figures5 and 6 for three different loading frequencies (5 Hz, 10 Hz, and 20 Hz). The transient response curves of displacement at track slab and roadbed show the form of sine plus carrier and change with the change of loading frequency and curve, of which the amplitude is very small. The vibration displacement amplitudes of track slab are 0.526 mm, 0.929 mm, and 0.993 mm at loading frequency of 5 Hz, 10 Hz, and 20 Hz. The vibration displacement amplitudes of roadbed are 0.053 mm, 0.057 mm, and 0.158 mm at loading frequency of 5 Hz, 10 Hz, and 20 Hz. The amplitudes of vibration displacement at two locations increase with the increase of loading frequency accordingly. Vibrations are found strongest at the track slab with a sharp decrease at the roadbed.Figure 5
Time history curves of dynamic displacement of track slab and roadbed for three loading frequencies. (a) Track slab (D1,f = 5 Hz, 10 Hz, and 20 Hz). (b) Roadbed (D2, f = 5 Hz, 10 Hz, and 20 Hz).
(a)(b)Figure 6
Frequency contents of vibration displacement at track slab (D1) and roadbed (D2). (a) Track slab (D1,f = 5 Hz, 10 Hz, and 20 Hz). (b) Roadbed (D2, f = 5 Hz, 10 Hz, 20 Hz).
(a)(b)In Figure6, the frequency contents are found mainly below 30 Hz. Information on the frequency contents shows peaks at 5 Hz, 10 Hz, and 20 Hz at loading frequency of 5 Hz, 10 Hz, and 20 Hz very clearly. The peak value of the displacement spectrum curve of track slab and roadbed reflects the frequency character of dynamic loading with sine wave. With the increase of loading frequency, the peak frequency spectrum also increases. The vibration displacement frequency characteristics reflect the accurate loading frequency characteristics of the track slab and roadbed. The roadbed has a certain characteristic frequency in the range of 0–30 Hz. Taking a loading frequency of 5 Hz, for example, in Figure 6(a), the main characteristic frequencies are 5 Hz, 10 Hz, 20 Hz, 25 Hz, and 30 Hz, respectively, below 30 Hz. There is a harmonic phenomenon, which is related to the vibration frequency character parameters of track slab and roadbed in low and medium frequency ranges (0–30 Hz). Dynamic response analysis under higher loading frequency (f > 20 Hz) is not considered in this paper.


### 3.2. Dynamic Velocity Response

Time histories and frequency contents of vibration velocity of track slab (V1), roadbed (V2), embankment (V3), and slope toe (V4) measured in the physical model are plotted and shown in Figures7 and 8 for three different loading frequencies (f = 5 Hz, 10 Hz, and 20 Hz). The positive and negative values represent the downward and upward vibration velocities, respectively. The time history curves of vibration velocity at each position present a sine wave form consistent with loading waves. The vibration characteristics of loading waves can be identified easily from the peaks and troughs in the dynamic response of velocity at many locations of track, embankment, and piled raft foundation. In Figure 7(a), at f = 5 Hz, the peak velocities of track slab, roadbed, and embankment are 16.1 mm/s, 1.37 mm/s, and 0.318 mm/s, which decrease along with the distance from the track center. In Figure 7(b), at f = 10 Hz, the peak velocity at track slab V1, roadbed V2, embankment V3, and slope toe V4 is 53.1 mm/s, 5.0 mm/s, 3.58 mm/s, and 3.57 mm/s. In Figure 7(c), at f = 10 Hz, the peak velocity at track slab V1, roadbed V2, embankment V3, and slope toe V4 is 80.8 mm/s, 24.9 mm/s, 14.27 mm/s, and 16.19 mm/s. The vibration velocities of the track slab and the embankment present an almost monotonically increasing tendency with the loading frequencies. This phenomenon can be explained by the frequency spectrum analysis shown in Figure 8.Figure 7
Time history curves of vibration velocity at track slab, roadbed, embankment, and slope toe. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)Figure 8
Frequency spectrum curves of vibration velocity at track slab, roadbed, embankment. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)In Figure8, the frequency contents and peak values are found mainly below 30 Hz. In Figure 8(a), with the loading frequency f = 5 Hz, information on the frequency contents shows peaks mainly at about 5 Hz, 10 Hz, 15 Hz, 20 Hz, 25 Hz, and 30 Hz and their related high-order harmonic frequencies. This phenomenon can be found also at other loading frequencies such as 10 Hz, 15 Hz, and 20 Hz. These dominant frequencies are proportionate to the increase of loading frequency, where the peak values occur naturally and have the maximum values at loading frequency points, larger than the others at high-order harmonic frequencies. Similar results can also be found in Bian et al.’s research [11]. These dominant frequencies are proportionate to the increasing train speed, which means they come from the loads caused by certain components of the train.Time histories and frequency contents of vibration velocity at the embankment (V6, V7), cushion (V8), and subsoil (V9, V10) measured in the physical model are plotted and shown in Figures9 and 10 for three different loading frequencies (f = 5 Hz, 10 Hz, 20 Hz). Frequency characteristics of sine wave loading can be shown by the velocity vibration response of velocity along with the depth from the roadbed surface. In Figure 9(a), the maximum values of velocity at V6, V7, V8, V9, and V10 are 1.48 mm/s, 0.44 mm/s, 0.25 mm/s, 0.08 mm/s, and 0.05 mm/s, respectively the velocity response on the track slab is much larger than that at other locations. In Figures 9(b) and 9(c), at f = 10 Hz, the maximum values of velocity at V6, V7, V8, V9, and V10 are 4.49 mm/s, 2.25 mm/s, 0.78 mm/s, 0.24 mm/s, and 0.2 mm/s. In Figure 9(c), at f = 20 Hz, the maximum values of velocity at V6, V7, V8, V9, and V10 are 9.7 mm/s, 7.06 mm/s, 2.62 mm/s, 1.67 mm/s, and 0.9 mm/s. The velocities at the track slab and embankment are larger than at the cushion and subsoil. The vibration velocity at the internal of track embankment is mostly smaller than that at the surface. With the increase of loading frequency, the vibration responses of the track structure and embankment have higher increasing rates than those of the substructure (raft, cushion, and subsoil). Such phenomena may be attributed to the different vibration characteristics of the track structure, piled raft foundation, and subsoils under dynamic loading frequencies. The structural damping, barrier, and weakening effect of piled raft and cushion may cause lower vibration in soft soil with higher material viscoelastic and damping.Figure 9
Time histories of vibration velocity at embankment, cushion, and subsoil for three cyclic loading frequencies. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)Figure 10
Frequency spectrum curves of vibration velocity at embankment, cushion, subsoil for three cyclic loading frequencies. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)In Figure10, for different substructures (roadbed, embankment, cushion, and soft soil), the frequency contents and peak values are found mainly corresponding to the loading frequency. In Figures 10(a)–10(c), for different loading frequencies, peak values can be found at the dominant frequencies and high-order harmonic frequencies. For example, f = 5 Hz, the dominant frequency is 5 Hz, and the high-order harmonic frequencies are 5 Hz, 15 Hz, 20 Hz, 25 Hz, and 30 Hz. The peaks at the dominant frequency are larger than these at the high-order harmonic frequency. The peaks at high-order harmonic frequency decrease with the increase of frequency.Figure11 shows the surface vibration velocity of track slab, roadbed, embankment, and raft in the horizontal direction from the track center under different loading frequencies. The vibration velocity amplitudes increase correspondingly with the increase of loading frequency. The vibration velocity amplitudes at track slab (V1) are 16.1 mm/s, 53.1 mm/s, and 80.8 mm/s at frequency of 5 Hz, 10 Hz, and 20 Hz, which reduce by about 91.5%, 90.6%, and 69.1% at the roadbed, respectively. Vibrations are found very large at the track slab with a sharp decrease at the roadbed, and higher frequency loading results in slower attenuation of the vibration velocities. The amplitude of vibration velocity at the roadbed surface reaches 0.063 m/s when the loading frequency increases to 20 Hz. Such strong vibrations in the roadbed will jeopardize the stability of the substructures with coarse granular materials. With the propagation of the vibrations, the vibration velocity amplitude decreases at a slower rate from the roadbed to the slope toe.Figure 11
Distribution of vibration velocity in the horizontal direction from the track center (f = 5 Hz, 10 Hz, 20 Hz).Figure12 shows the distribution of vibration velocity in the vertical direction along with the depth from the roadbed surface in the middle cross section. The vibration velocity amplitudes at the top of the embankment (V6) at 0.15 m depth from the roadbed surface are 1.48 mm/s, 4.49 mm/s, and 9.7 mm/s at the frequency of 5 Hz, 10 Hz, and 20 Hz, which reduce by about 70.5%, 50%, and 27.2% at the bottom of embankment (V7) at 0.54 m depth from the roadbed surface, respectively, while the decrease rates are 82.8%, 82.6%, and 73% at cushion (V8) at 0.66 m depth from the roadbed surface. The decrease rates at subsoil 2.87 m depth from the roadbed surface are 94.3%, 94.7%, and 82.8%. Vibration velocity decreases at much faster rates within 0.66 m depth from the roadbed surface, and then the rates slow down. With the increase of loading frequency, the vibration responses of the embankment have higher increasing rates than those of raft, cushion, and subsoil. The piled raft has an obvious influence on the vibration velocity attenuation with the increase of loading frequency within 0.66 m depth from the roadbed surface. With the increase of depth, the difference of vibration velocity attenuation rate decreases gradually.Figure 12
Distribution of vibration velocity in the vertical direction along with the depth from the roadbed surface.Based on the safety considerations of test equipment and personnel, the frequency range of cyclic loading sine wave is controlled below 20 Hz to obtain the peak changes of vibration velocity at different frequencies. Figure13 shows the relation between the vibration velocity and loading frequency in the model test. The vibration velocity of the track slab, roadbed, embankment, and slope toe increases with the loading frequency. For cyclic sine wave loading, the vibration velocity of the track slab varies clearly with frequency and increases rapidly. As is shown, the vibration velocity of the track slab increases gradually and linearly with cyclic frequency below 5 Hz, which has a higher increasing range and rate than those of roadbed, embankment, and slope toe in the frequency range 0-9 Hz. Vibration velocity of the track slab increases fast between 5 and 9 Hz and then slows down, increasing with the increase of frequency below 12 Hz. The rate of increment becomes rapid between 12 and 15 Hz and then slows down after 15 Hz, which has a lower increasing range and rate than those at roadbed, embankment, and slope toe in the frequency range 15–20 Hz.Figure 13
Relation between surface vibration velocity and loading frequency in the model test.


### 3.3. Dynamic Stress Response

Figure14 shows the time histories of vertical dynamic soil stresses at different depth from the roadbed surface (E1, E2, E3, and E6, shown in Figure 2). It can be seen that the vibration response of the substructure of the track under sine wave loading is obvious. In the substructure of track-embankment-piled raft foundation, the variation of dynamic soil stress caused by cyclic sine wave loading shows the characteristics of sine waves corresponding to loading waves at different frequencies. The positive and negative values of stress vibration curves represent the downward and upward vibrations.Figure 14
Time history analysis of dynamic soil stress at embankment, cushion, and subsoil for three cyclic loading frequencies. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)In Figure14(a), at f = 5 Hz, the maximum dynamic soil stresses at E1, E2, E3, and E6 are 0.733 kPa, 0.271 kPa, 0.263 kPa, and 0.059 kPa. In Figure 14(b), at f = 10 Hz, the maximum dynamic soil stresses at E1, E2, E3, and E6 are 0.99 kPa, 0.74 kPa, 0.42 kPa, and 0.075 kPa. In Figure 14(c), at f = 20 Hz, the maximum dynamic soil stresses at E1, E2, E3, and E6 are 2.65 kPa, 1.8 kPa, 1.23 kPa, and 0.41 kPa. With the increase of loading frequency, the dynamic stresses at different locations increase correspondingly in different degrees. The peak value of vibration stress caused by the sine loading wave is clearly visible, and the vibration pattern of vibration soil stress reflects the fluctuation law of cyclic sine wave loading. With the depth increase, the attenuation rate of dynamic soil stresses decreases from the embankment surface to the bottom of the soft subsoil, reaching 92%, 92.5%, and 84.5% for loading frequency 5 Hz, 10 Hz, and 20 Hz. The piled raft and cushion structures have vibration attenuation effect on the transmission of vibration stress.In Figures15(a)–15(c), for different substructures (embankment, cushion, and subsoil), the frequency contents and peak values are found mainly corresponding to the loading frequency. The peak values can be found at the dominant frequencies and high-order harmonic frequencies, which are similar to the frequency pattern of vibration velocity in Figures 8 and 10. The peaks at the dominant frequency are larger than these at the high-order harmonic frequency. The peaks at high-order harmonic frequency decrease with the increase of frequency.Figure 15
Frequency spectral analysis of dynamic soil stress at embankment, cushion, and subsoil for three cyclic loading frequencies. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)Figure16 shows the distribution of dynamic soil stress along with the depth from the roadbed surface in the middle cross section. It is found in Figure 14 that, with the increase of loading frequency, the dynamic stresses at different locations increase correspondingly in different degrees. The dynamic soil stress amplitudes at the top of the embankment (E1) at 0.15 m depth from the roadbed surface are 0.73 kPa, 0.989 kPa, and 2.65 kPa at the frequency of 5 Hz, 10 Hz, and 20 Hz, which reduce by about 63%, 24.9%, and 32% at the bottom of embankment (E2) at 0.54 m depth from the roadbed surface, respectively, while the decrease rates are 64.1%, 58%, and 53.5% at cushion (E3) at 0.66 m depth from the roadbed surface. The decrease rates at subsoil (E6) at 5.02 m depth from the roadbed surface are 92%, 92.4%, and 84.5%. Dynamic soil stress decreases at a much faster rate within 0.66 m depth from the roadbed surface, and then the rate slows down at the subsoil. The attenuation of dynamic soil stress is caused by vibration damping attenuation of soft soil foundation. In the soft soil foundation, the increase rate and level of stress are much lower than those in track, embankment, and piled raft structures along with the soil depth from the surface to subsoil, especially in the underlying soil layers.Figure 16
Distribution of dynamic soil stresses in the track substructure (embankment, raft, cushion, and subsoil).Figure17 shows the dynamic soil stress amplitudes at various locations E1, E2, E3, and E6. As is shown in Figure 14, with the increase of loading frequency, the dynamic stresses at different locations increase correspondingly in different degrees. In Figure 17, the amplitude of dynamic soil stresses at embankment (E1, E2) and cushion (E3) is sensitive to the loading frequency; they show an obvious trend with the increase of loading frequency. The dynamic soil stress amplitudes generally exhibit slow ascendant tendencies when the loading frequency is lower than 5 Hz, but these values grow quickly with the loading frequency in the range from 5 Hz to about 20 Hz and have some fluctuations in the range from 9 Hz to 11 Hz. The increase rate and amplitude at the top of the embankment (E1) near the track slab are larger than at the deeper place of embankment (E2), cushion (E3), and subsoil (E6), while for the deeper locations of the subsoil (E6), the situation is different. The amplitude of dynamic soil stresses increases slowly with the increase of loading frequency, which has some improvement in the increase rate when the loading frequency is higher than 15 Hz, where the frequency is close to the subsoil’s resonance resonant period frequency.Figure 17
Relation between dynamic soil stress and loading frequency in the track substructure.


### 3.4. Pile Stress Response Analysis

The time histories and frequency analysis of dynamic stress at the pile top and bottom forf = 20 Hz are shown in Figure 18. The dynamic loads are induced by vibration exciter, which transfer stresses from track slab to roadbed, embankment, raft, cushion, piles, and subsoils. The dynamic response of stress caused by the load transferred to the pile top exhibits the shape and frequency characteristic of the loading sine wave. The positive and negative dynamic stress responses of the concrete pile represent the repeated tensile and compressive stresses in pile. Dynamic stress propagates downward along the pile and decreases gradually from pile top 4.3 kPa to pile bottom 0.63 kPa, of which the attenuation rate reaches 85.3%. Compared with the dynamic stress at the pile top, cushion, subsoils, and pile bottom in Figures 14(a) and 18, for the loading frequency of 20 Hz, the maximum dynamic soil stresses at S1, E3, E6, and S3 are 4.3 kPa, 1.23 kPa, 0.41 kPa, and 0.63 kPa. The load sharing ratio between pile and gravel cushion is 3.5, which is similar to the load sharing and transfer mechanism under static load for X-section piled raft foundation. The pile bears more load than cushion and subsoils.Figure 18
Time history analysis of dynamic stress at pile top and bottom forf = 20 Hz.The frequency spectral analysis curves of dynamic stress at the pile top and bottom forf = 20 Hz are shown in Figure 19 under sine wave cyclic loading. The dominant and harmonic frequencies of the dynamic stress of the pile are 20 Hz, 40 Hz, 60 Hz, and 80 Hz and are distributed in the middle and low frequency ranges, which is related to the vibration stress response and material characteristics of pile.Figure 19
Frequency spectral analysis of dynamic stress at pile top and bottom forf = 20 Hz.


## 3.1. Dynamic Displacement Response

Cyclic axial loading on the track slab resulted in the dynamic displacement of the track slab and underlying structures. Time histories and frequency contents of the vibration displacement of track slab (D1) and roadbed (D2) measured in the physical model are plotted and shown in Figures5 and 6 for three different loading frequencies (5 Hz, 10 Hz, and 20 Hz). The transient response curves of displacement at track slab and roadbed show the form of sine plus carrier and change with the change of loading frequency and curve, of which the amplitude is very small. The vibration displacement amplitudes of track slab are 0.526 mm, 0.929 mm, and 0.993 mm at loading frequency of 5 Hz, 10 Hz, and 20 Hz. The vibration displacement amplitudes of roadbed are 0.053 mm, 0.057 mm, and 0.158 mm at loading frequency of 5 Hz, 10 Hz, and 20 Hz. The amplitudes of vibration displacement at two locations increase with the increase of loading frequency accordingly. Vibrations are found strongest at the track slab with a sharp decrease at the roadbed.Figure 5
Time history curves of dynamic displacement of track slab and roadbed for three loading frequencies. (a) Track slab (D1,f = 5 Hz, 10 Hz, and 20 Hz). (b) Roadbed (D2, f = 5 Hz, 10 Hz, and 20 Hz).
(a)(b)Figure 6
Frequency contents of vibration displacement at track slab (D1) and roadbed (D2). (a) Track slab (D1,f = 5 Hz, 10 Hz, and 20 Hz). (b) Roadbed (D2, f = 5 Hz, 10 Hz, 20 Hz).
(a)(b)In Figure6, the frequency contents are found mainly below 30 Hz. Information on the frequency contents shows peaks at 5 Hz, 10 Hz, and 20 Hz at loading frequency of 5 Hz, 10 Hz, and 20 Hz very clearly. The peak value of the displacement spectrum curve of track slab and roadbed reflects the frequency character of dynamic loading with sine wave. With the increase of loading frequency, the peak frequency spectrum also increases. The vibration displacement frequency characteristics reflect the accurate loading frequency characteristics of the track slab and roadbed. The roadbed has a certain characteristic frequency in the range of 0–30 Hz. Taking a loading frequency of 5 Hz, for example, in Figure 6(a), the main characteristic frequencies are 5 Hz, 10 Hz, 20 Hz, 25 Hz, and 30 Hz, respectively, below 30 Hz. There is a harmonic phenomenon, which is related to the vibration frequency character parameters of track slab and roadbed in low and medium frequency ranges (0–30 Hz). Dynamic response analysis under higher loading frequency (f > 20 Hz) is not considered in this paper.


## 3.2. Dynamic Velocity Response

Time histories and frequency contents of vibration velocity of track slab (V1), roadbed (V2), embankment (V3), and slope toe (V4) measured in the physical model are plotted and shown in Figures7 and 8 for three different loading frequencies (f = 5 Hz, 10 Hz, and 20 Hz). The positive and negative values represent the downward and upward vibration velocities, respectively. The time history curves of vibration velocity at each position present a sine wave form consistent with loading waves. The vibration characteristics of loading waves can be identified easily from the peaks and troughs in the dynamic response of velocity at many locations of track, embankment, and piled raft foundation. In Figure 7(a), at f = 5 Hz, the peak velocities of track slab, roadbed, and embankment are 16.1 mm/s, 1.37 mm/s, and 0.318 mm/s, which decrease along with the distance from the track center. In Figure 7(b), at f = 10 Hz, the peak velocity at track slab V1, roadbed V2, embankment V3, and slope toe V4 is 53.1 mm/s, 5.0 mm/s, 3.58 mm/s, and 3.57 mm/s. In Figure 7(c), at f = 10 Hz, the peak velocity at track slab V1, roadbed V2, embankment V3, and slope toe V4 is 80.8 mm/s, 24.9 mm/s, 14.27 mm/s, and 16.19 mm/s. The vibration velocities of the track slab and the embankment present an almost monotonically increasing tendency with the loading frequencies. This phenomenon can be explained by the frequency spectrum analysis shown in Figure 8.Figure 7
Time history curves of vibration velocity at track slab, roadbed, embankment, and slope toe. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)Figure 8
Frequency spectrum curves of vibration velocity at track slab, roadbed, embankment. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)In Figure8, the frequency contents and peak values are found mainly below 30 Hz. In Figure 8(a), with the loading frequency f = 5 Hz, information on the frequency contents shows peaks mainly at about 5 Hz, 10 Hz, 15 Hz, 20 Hz, 25 Hz, and 30 Hz and their related high-order harmonic frequencies. This phenomenon can be found also at other loading frequencies such as 10 Hz, 15 Hz, and 20 Hz. These dominant frequencies are proportionate to the increase of loading frequency, where the peak values occur naturally and have the maximum values at loading frequency points, larger than the others at high-order harmonic frequencies. Similar results can also be found in Bian et al.’s research [11]. These dominant frequencies are proportionate to the increasing train speed, which means they come from the loads caused by certain components of the train.Time histories and frequency contents of vibration velocity at the embankment (V6, V7), cushion (V8), and subsoil (V9, V10) measured in the physical model are plotted and shown in Figures9 and 10 for three different loading frequencies (f = 5 Hz, 10 Hz, 20 Hz). Frequency characteristics of sine wave loading can be shown by the velocity vibration response of velocity along with the depth from the roadbed surface. In Figure 9(a), the maximum values of velocity at V6, V7, V8, V9, and V10 are 1.48 mm/s, 0.44 mm/s, 0.25 mm/s, 0.08 mm/s, and 0.05 mm/s, respectively the velocity response on the track slab is much larger than that at other locations. In Figures 9(b) and 9(c), at f = 10 Hz, the maximum values of velocity at V6, V7, V8, V9, and V10 are 4.49 mm/s, 2.25 mm/s, 0.78 mm/s, 0.24 mm/s, and 0.2 mm/s. In Figure 9(c), at f = 20 Hz, the maximum values of velocity at V6, V7, V8, V9, and V10 are 9.7 mm/s, 7.06 mm/s, 2.62 mm/s, 1.67 mm/s, and 0.9 mm/s. The velocities at the track slab and embankment are larger than at the cushion and subsoil. The vibration velocity at the internal of track embankment is mostly smaller than that at the surface. With the increase of loading frequency, the vibration responses of the track structure and embankment have higher increasing rates than those of the substructure (raft, cushion, and subsoil). Such phenomena may be attributed to the different vibration characteristics of the track structure, piled raft foundation, and subsoils under dynamic loading frequencies. The structural damping, barrier, and weakening effect of piled raft and cushion may cause lower vibration in soft soil with higher material viscoelastic and damping.Figure 9
Time histories of vibration velocity at embankment, cushion, and subsoil for three cyclic loading frequencies. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)Figure 10
Frequency spectrum curves of vibration velocity at embankment, cushion, subsoil for three cyclic loading frequencies. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)In Figure10, for different substructures (roadbed, embankment, cushion, and soft soil), the frequency contents and peak values are found mainly corresponding to the loading frequency. In Figures 10(a)–10(c), for different loading frequencies, peak values can be found at the dominant frequencies and high-order harmonic frequencies. For example, f = 5 Hz, the dominant frequency is 5 Hz, and the high-order harmonic frequencies are 5 Hz, 15 Hz, 20 Hz, 25 Hz, and 30 Hz. The peaks at the dominant frequency are larger than these at the high-order harmonic frequency. The peaks at high-order harmonic frequency decrease with the increase of frequency.Figure11 shows the surface vibration velocity of track slab, roadbed, embankment, and raft in the horizontal direction from the track center under different loading frequencies. The vibration velocity amplitudes increase correspondingly with the increase of loading frequency. The vibration velocity amplitudes at track slab (V1) are 16.1 mm/s, 53.1 mm/s, and 80.8 mm/s at frequency of 5 Hz, 10 Hz, and 20 Hz, which reduce by about 91.5%, 90.6%, and 69.1% at the roadbed, respectively. Vibrations are found very large at the track slab with a sharp decrease at the roadbed, and higher frequency loading results in slower attenuation of the vibration velocities. The amplitude of vibration velocity at the roadbed surface reaches 0.063 m/s when the loading frequency increases to 20 Hz. Such strong vibrations in the roadbed will jeopardize the stability of the substructures with coarse granular materials. With the propagation of the vibrations, the vibration velocity amplitude decreases at a slower rate from the roadbed to the slope toe.Figure 11
Distribution of vibration velocity in the horizontal direction from the track center (f = 5 Hz, 10 Hz, 20 Hz).Figure12 shows the distribution of vibration velocity in the vertical direction along with the depth from the roadbed surface in the middle cross section. The vibration velocity amplitudes at the top of the embankment (V6) at 0.15 m depth from the roadbed surface are 1.48 mm/s, 4.49 mm/s, and 9.7 mm/s at the frequency of 5 Hz, 10 Hz, and 20 Hz, which reduce by about 70.5%, 50%, and 27.2% at the bottom of embankment (V7) at 0.54 m depth from the roadbed surface, respectively, while the decrease rates are 82.8%, 82.6%, and 73% at cushion (V8) at 0.66 m depth from the roadbed surface. The decrease rates at subsoil 2.87 m depth from the roadbed surface are 94.3%, 94.7%, and 82.8%. Vibration velocity decreases at much faster rates within 0.66 m depth from the roadbed surface, and then the rates slow down. With the increase of loading frequency, the vibration responses of the embankment have higher increasing rates than those of raft, cushion, and subsoil. The piled raft has an obvious influence on the vibration velocity attenuation with the increase of loading frequency within 0.66 m depth from the roadbed surface. With the increase of depth, the difference of vibration velocity attenuation rate decreases gradually.Figure 12
Distribution of vibration velocity in the vertical direction along with the depth from the roadbed surface.Based on the safety considerations of test equipment and personnel, the frequency range of cyclic loading sine wave is controlled below 20 Hz to obtain the peak changes of vibration velocity at different frequencies. Figure13 shows the relation between the vibration velocity and loading frequency in the model test. The vibration velocity of the track slab, roadbed, embankment, and slope toe increases with the loading frequency. For cyclic sine wave loading, the vibration velocity of the track slab varies clearly with frequency and increases rapidly. As is shown, the vibration velocity of the track slab increases gradually and linearly with cyclic frequency below 5 Hz, which has a higher increasing range and rate than those of roadbed, embankment, and slope toe in the frequency range 0-9 Hz. Vibration velocity of the track slab increases fast between 5 and 9 Hz and then slows down, increasing with the increase of frequency below 12 Hz. The rate of increment becomes rapid between 12 and 15 Hz and then slows down after 15 Hz, which has a lower increasing range and rate than those at roadbed, embankment, and slope toe in the frequency range 15–20 Hz.Figure 13
Relation between surface vibration velocity and loading frequency in the model test.


## 3.3. Dynamic Stress Response

Figure14 shows the time histories of vertical dynamic soil stresses at different depth from the roadbed surface (E1, E2, E3, and E6, shown in Figure 2). It can be seen that the vibration response of the substructure of the track under sine wave loading is obvious. In the substructure of track-embankment-piled raft foundation, the variation of dynamic soil stress caused by cyclic sine wave loading shows the characteristics of sine waves corresponding to loading waves at different frequencies. The positive and negative values of stress vibration curves represent the downward and upward vibrations.Figure 14
Time history analysis of dynamic soil stress at embankment, cushion, and subsoil for three cyclic loading frequencies. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)In Figure14(a), at f = 5 Hz, the maximum dynamic soil stresses at E1, E2, E3, and E6 are 0.733 kPa, 0.271 kPa, 0.263 kPa, and 0.059 kPa. In Figure 14(b), at f = 10 Hz, the maximum dynamic soil stresses at E1, E2, E3, and E6 are 0.99 kPa, 0.74 kPa, 0.42 kPa, and 0.075 kPa. In Figure 14(c), at f = 20 Hz, the maximum dynamic soil stresses at E1, E2, E3, and E6 are 2.65 kPa, 1.8 kPa, 1.23 kPa, and 0.41 kPa. With the increase of loading frequency, the dynamic stresses at different locations increase correspondingly in different degrees. The peak value of vibration stress caused by the sine loading wave is clearly visible, and the vibration pattern of vibration soil stress reflects the fluctuation law of cyclic sine wave loading. With the depth increase, the attenuation rate of dynamic soil stresses decreases from the embankment surface to the bottom of the soft subsoil, reaching 92%, 92.5%, and 84.5% for loading frequency 5 Hz, 10 Hz, and 20 Hz. The piled raft and cushion structures have vibration attenuation effect on the transmission of vibration stress.In Figures15(a)–15(c), for different substructures (embankment, cushion, and subsoil), the frequency contents and peak values are found mainly corresponding to the loading frequency. The peak values can be found at the dominant frequencies and high-order harmonic frequencies, which are similar to the frequency pattern of vibration velocity in Figures 8 and 10. The peaks at the dominant frequency are larger than these at the high-order harmonic frequency. The peaks at high-order harmonic frequency decrease with the increase of frequency.Figure 15
Frequency spectral analysis of dynamic soil stress at embankment, cushion, and subsoil for three cyclic loading frequencies. (a)f = 5 Hz. (b) f = 10 Hz. (c) f = 20 Hz.
(a)(b)(c)Figure16 shows the distribution of dynamic soil stress along with the depth from the roadbed surface in the middle cross section. It is found in Figure 14 that, with the increase of loading frequency, the dynamic stresses at different locations increase correspondingly in different degrees. The dynamic soil stress amplitudes at the top of the embankment (E1) at 0.15 m depth from the roadbed surface are 0.73 kPa, 0.989 kPa, and 2.65 kPa at the frequency of 5 Hz, 10 Hz, and 20 Hz, which reduce by about 63%, 24.9%, and 32% at the bottom of embankment (E2) at 0.54 m depth from the roadbed surface, respectively, while the decrease rates are 64.1%, 58%, and 53.5% at cushion (E3) at 0.66 m depth from the roadbed surface. The decrease rates at subsoil (E6) at 5.02 m depth from the roadbed surface are 92%, 92.4%, and 84.5%. Dynamic soil stress decreases at a much faster rate within 0.66 m depth from the roadbed surface, and then the rate slows down at the subsoil. The attenuation of dynamic soil stress is caused by vibration damping attenuation of soft soil foundation. In the soft soil foundation, the increase rate and level of stress are much lower than those in track, embankment, and piled raft structures along with the soil depth from the surface to subsoil, especially in the underlying soil layers.Figure 16
Distribution of dynamic soil stresses in the track substructure (embankment, raft, cushion, and subsoil).Figure17 shows the dynamic soil stress amplitudes at various locations E1, E2, E3, and E6. As is shown in Figure 14, with the increase of loading frequency, the dynamic stresses at different locations increase correspondingly in different degrees. In Figure 17, the amplitude of dynamic soil stresses at embankment (E1, E2) and cushion (E3) is sensitive to the loading frequency; they show an obvious trend with the increase of loading frequency. The dynamic soil stress amplitudes generally exhibit slow ascendant tendencies when the loading frequency is lower than 5 Hz, but these values grow quickly with the loading frequency in the range from 5 Hz to about 20 Hz and have some fluctuations in the range from 9 Hz to 11 Hz. The increase rate and amplitude at the top of the embankment (E1) near the track slab are larger than at the deeper place of embankment (E2), cushion (E3), and subsoil (E6), while for the deeper locations of the subsoil (E6), the situation is different. The amplitude of dynamic soil stresses increases slowly with the increase of loading frequency, which has some improvement in the increase rate when the loading frequency is higher than 15 Hz, where the frequency is close to the subsoil’s resonance resonant period frequency.Figure 17
Relation between dynamic soil stress and loading frequency in the track substructure.


## 3.4. Pile Stress Response Analysis

The time histories and frequency analysis of dynamic stress at the pile top and bottom forf = 20 Hz are shown in Figure 18. The dynamic loads are induced by vibration exciter, which transfer stresses from track slab to roadbed, embankment, raft, cushion, piles, and subsoils. The dynamic response of stress caused by the load transferred to the pile top exhibits the shape and frequency characteristic of the loading sine wave. The positive and negative dynamic stress responses of the concrete pile represent the repeated tensile and compressive stresses in pile. Dynamic stress propagates downward along the pile and decreases gradually from pile top 4.3 kPa to pile bottom 0.63 kPa, of which the attenuation rate reaches 85.3%. Compared with the dynamic stress at the pile top, cushion, subsoils, and pile bottom in Figures 14(a) and 18, for the loading frequency of 20 Hz, the maximum dynamic soil stresses at S1, E3, E6, and S3 are 4.3 kPa, 1.23 kPa, 0.41 kPa, and 0.63 kPa. The load sharing ratio between pile and gravel cushion is 3.5, which is similar to the load sharing and transfer mechanism under static load for X-section piled raft foundation. The pile bears more load than cushion and subsoils.Figure 18
Time history analysis of dynamic stress at pile top and bottom forf = 20 Hz.The frequency spectral analysis curves of dynamic stress at the pile top and bottom forf = 20 Hz are shown in Figure 19 under sine wave cyclic loading. The dominant and harmonic frequencies of the dynamic stress of the pile are 20 Hz, 40 Hz, 60 Hz, and 80 Hz and are distributed in the middle and low frequency ranges, which is related to the vibration stress response and material characteristics of pile.Figure 19
Frequency spectral analysis of dynamic stress at pile top and bottom forf = 20 Hz.


## 4. Conclusions

Large-scale model tests were performed to investigate the dynamic response and bear capacity ofX-section piled raft foundation under cyclic axial load in soft soil. The main conclusions drawn from the testing results presented in this paper are as follows:(1)
The vibration displacement, velocity, and stress of the track, embankment, and piled raft composite foundation follow a pattern of vibration characteristics of loading sine wave. The vibration characteristics of loading waves can be identified easily from the peaks and troughs in the dynamic response of velocity at many locations of track, embankment, and piled raft foundation. With the increase of loading frequency, the vibration responses of the track structure and embankment have higher increasing rates than those at the substructure (raft, cushion, and subsoil). Vibrations are found very large at the track slab with sharp decrease at the roadbed, and higher frequency loading results in slower attenuation of the vibration velocities. Vibration velocity decreases at much faster rates within 0.66 m depth from the roadbed surface, and then the rates slow down. The dominant frequencies of frequency contents are proportionate to the increase of loading frequency, where the peak values occur naturally and have a maximum value at loading frequency points, larger than others at high-order harmonic frequencies.(2)
Dynamic soil stress decreases at much faster rates within 0.66 m depth from the roadbed surface, and then the rates slow down at the subsoil. The piled raft and cushion structures have vibration attenuation effect on the transmission of vibration stress. The attenuation rate of dynamic soil stresses decreases from the embankment surface to bottom of soft subsoil, reaching 92%, 92.5%, and 84.5% for loading frequency 5 Hz, 10 Hz, and 20 Hz. The attenuation of dynamic soil stress is caused by vibration damping attenuation of soft soil foundation.(3)
The dynamic response of stress caused by the load transferred to the pile top exhibits the shape and frequency characteristic of the loading sine wave. The load sharing ratio between pile and gravel cushion is 3.5, which is similar to the load sharing and transfer mechanism under static load forX-section piled raft foundation. This result indicates that the piled raft structure bears more dynamic load than cushion and subsoils, to ensure long-term dynamic stability and safety of soft soil foundation for ballastless track.



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*Source: 4561806-2021-10-19.xml*