1 Introduction

It is widely reported that a large number of earth embankments or slopes were damaged during the past great earthquakes (Omer and Resat 2002; Bakir and Akis 2005; Singh et al. 2005; Kamai and Sangawa 2011; Toyohiko et al. 2011; Huang et al. 2012a, 2013). In 2008, about 197,481 slope failures were triggered by Wenchuan earthquake of China (M w 7.9 or M s 8.0) (Xu and Xu 2013; Xu et al. 2014c), and the phenomena of seismic liquefaction and subsidence were also observed (Huang and Jiang 2010; Huang et al. 2011, 2012b, c). The Yushu, China, earthquake in 2010 (M w 6.9) triggered more than 2,036 co-seismic landslides with a total coverage area of 1.194 km2 (Xu and Xu 2014). The Port-au-Prince, Haiti earthquake in 2010 (M w7.0) induced more than 30,828 slope failures, which distributed in an area larger than 3,000 km2, and covered about 15.736 km2 with an estimated landslide erosion volume about 29,700,000 m3 (Xu et al. 2014a). The Minxian–Zhangxian, China, earthquake in 2013 (M w 5.9) triggered at least 2,330 slope failures (Xu et al. 2014b). The dynamic behavior of earth embankment or slope subjected to earthquake loading requires a further understanding for creating a safe seismic design.

Traditionally, the seismic stability of embankment or slope was analyzed by pseudostatic analysis method which simply added the seismic force to the static limit equilibrium analysis (Liu and Shi 2006; Shukha and Baker 2008; Munwar and Sivakumar 2009). The pseudostatic analysis method was widely used for seismic design in current codes for its briefness. The deformation control is another method for seismic design of embankment or slope. Newmark (1965) presented an analysis method by calculating the permanent displacement as a function of the critical acceleration, which was later well known as Newmark’s sliding block method. Newmark’s sliding block method is simple to apply in practice and it was greatly improved by many scholars (Chugh 1995; Crespellani et al. 2003; Hsieh and Lee 2011; Kim and Sitar 2013). However, the Newmark’s sliding block method still presented some of crude assumptions of pseudostatic analysis (Jibson 2011).

The rapid development of centrifuge technique and shaking table apparatus makes it possible to study the seismic response of earth embankment or slope under real stress–strain condition in laboratory. In recent years, dynamic centrifuge tests or shaking table tests were widely adopted by scholars to investigate the seismic response of geotechnical structures (Mohajeri and Towhata 2003; Teymur and Madabhushi 2003; Kagawa et al. 2004; Rayhani and El Naggar 2007; Torisu et al. 2010; Motamed et al. 2013; Srilatha et al. 2013; Turan et al. 2013). In dynamic centrifuge test, the test model can well meet the requirements of similarity relation because the gravitational field can be changed as expected. However, the carrying capacity of dynamic centrifuge is very limited due to the complicated technology of centrifuge system, and most of the existing literatures mainly performed dynamic centrifuge test on small-scale model (Yu et al. 2008, 2010; Ha et al. 2014). Instead, the shaking table system can support a large-scale model with a wide loading frequency in despite of some shortages in similarity design. Liu et al. (2014) performed a series of large-scale shaking table model tests to evaluate the effectiveness of various reinforcing measures on concrete faced rock-fill dams. The dimension of the model is 4.0 m × 0.9 m × 1.5 m (length × width × height). Wang et al. (2010) conducted shaking table model tests to study the failure behavior of slope with a model dimension of 3.5 m × 1.5 m × 1.7 m under Wenchuan excitation with a conclusion that the tension cracks mainly emerged at the upper part of slope, while the bottom of slope nearly remained intact.

The seismic stability or engineering behavior of earth embankment or slope can be well improved by taking reinforcing measure, which is investigated both in field survey and laboratory test (Tatsuoka et al. 2007; Wang et al. 2011; Liu et al. 2012; Huang et al. 2006). For example, Sandri (1997) presented a summary of observations on retaining walls and reinforced soil slopes after Northridge earthquake, and the results showed that the reinforced structures performed excellently subjected to earthquake loading. Hong et al. (2005) performed shaking table tests on five-nailed model slopes, and the research showed that the nails could improve the seismic resistance of steep slopes notably. The previously published works promoted the application of reinforcing measure and developed the methods for seismic design of reinforced structure. However, an economic and effective reinforcing measure requires a comprehensive understanding on seismic response of different patterns of reinforced embankment or slope, which should be further studied.

In this work, large-scale shaking table model tests were carried out on three embankment slope models with different reinforcing measures (i.e., unreinforced embankment slope, 2-layer reinforced embankment slope and 4-layer reinforced embankment slope). Wenchuan earthquake motions and white noise excitations were mainly performed in the test to investigate the seismic response of embankment slopes including the change of model parameter, the horizontal acceleration response, the vertical acceleration response and the dynamic earth pressure response. The seismic response among the embankment slopes with different reinforcing measures was also compared.

2 Shaking table tests

2.1 Similarity laws and materials

Shaking table tests were performed on large-scale shaking table established at China Merchants Chongqing Communications Research and Design Institute Co. Ltd, which was equipped with electro-hydraulic servo-controlled system. The dimension of the table is 6 m × 3 m (length × width). The shaking table system can support the test model with the weight of 350 kN and provide a loading frequency of 0.1–50 Hz.

Shaking table test model was designed based on the prototype of single-line unballasted track railway embankment slope, which was constructed on rock foundation. According to the dimension of prototype and the capacity of shaking table system, the similarity ratio of geometry size was determined to be 1:8. Taking the parameters including geometry size, mass density and gravity acceleration as control variables, the similarity laws for embankment slope can be deduced based on Buckingham π theory with the assumption that the soil obeys Mohr–Coulomb criterion in static condition and equivalent viscoelastic criterion in dynamic condition (Iai 1989; Lin and Wang 2006; Wang et al. 2010; Lin 2013a, b; Lin and Yang 2013). The similarity laws of embankment slope are shown in Table 1. In which, input acceleration and time were set as input control variables in shaking table test.

Table 1 Similarity laws of embankment slope in shaking table test
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The soil used in shaking table tests was obtained from a deposit in Chongqing city, China. Based on grading analysis test, the non-uniform coefficient and curvature coefficient of the soil were 11.5 and 1.04, respectively. The soil was defined as “A-filler” according to the “Code for Design on Subgrade of Railway” (GB10001-2005) in China. The cohesion and internal friction angle of the soil were 11.2 kPa and 24.3°. The optimum water content and the maximum dry unit weight were 8.44 % and 20.2 kN/m3. Bi-directional geogrid of TGSG25-25 was used in reinforced embankment slopes. According to tensile test, the longitudinal and transverse tensile strengths of the geogrid were 26.0 and 25.1 kN/m. The longitudinal and transverse elongations were 15.0 and 12.0 %, respectively.

2.2 Test cases and input excitations

Three groups of shaking table test were performed on embankment slopes with different reinforcing measures, and they were code-named unreinforced embankment slope, 2-layer reinforced embankment slope and 4-layer reinforced embankment slope, respectively. Every embankment slope model was constructed in a special model box with a dimension of 3.6 m × 1.5 m × 2.0 m (length × width × height). According to the similarity laws of embankment slope, rock foundation was simulated by concrete layer with a thickness of 0.4 m at the bottom of model box. The embankment body and bottom layer were designed to be 1.2 and 0.3 m in height, respectively. The graded gravel board was determined to be 0.1 m in height with a consideration of the counterweight of upper structure on railway embankment, as shown in Fig. 1. The compaction degrees of embankment body and bottom layer were 87 and 95 %, respectively. In 2-layer reinforced embankment slope, two pieces of TGSG25-25 geogrid were laid in the elevations of 0.5 and 1.0 m to rock foundation. In 4-layer reinforced embankment slope, four pieces of TGSG25-25 geogrid were laid in the elevations of 0.3, 0.6, 0.9 and 1.0 m to rock foundation. The length of geogrid was all designed to be 1.0 m, as shown in Table 2. A photograph of embankment slope model on shaking table is shown in Fig. 2.

Fig. 1
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Embankment slope model and instruments layout (unit: m)

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Table 2 Summary of different patterns of reinforced embankment slopes
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Fig. 2
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An embankment slope model on shaking table

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In order to reduce the “box-effect” in shaking table test, the natural frequency of the model box should not be close to that of the soil. In this paper, the model box was diagonally supported by U-steel on two sides to improve the stiffness of the model box, which subsequently increased the natural frequency of the model box. The results showed that the natural frequency of the model box reached 120 Hz, which was much larger than that of the soil. Polystyrene foam board was set inside the model box to reduce the wave reflection during the test. More details about the model box, the soil and the embankment model were described by Lin and Yang (2013).

The characteristic of near-field ground motion is greatly different from that of far-field motion, and the structure will be more vulnerable subjected to near-field ground motion (Alavi and Krawinkler 2004; Chopra and Chintanapakdee 2001a, b; Yaghmaei-Sabegha and Tsang 2011; Davoodi et al. 2013). Consequently, Wenchuan earthquake motion, which was the nearest ground motion data recorded at Wolong station in Wenchuan earthquake in 2008, was selected for test. Wenchuan earthquake motion was a long-period pulse with two great impulsions. The timeline of Wenchuan earthquake motion was compressed according to the time similarity constant (C t = 4.76) in shaking table test. The time history and the Fourier spectrum of compressed Wenchuan earthquake motion are shown in Fig. 3. The Wenchuan earthquake motion was performed in bi-direction during the test. The horizontal peak acceleration of Wenchuan excitation was adjusted to 0.1, 0.2, 0.4, 0.6, 0.8 and 1.0 g, respectively. Correspondingly, the vertical peak acceleration was adjusted to 2/3 of horizontal peak acceleration for each Wechuan excitation. White noises with peak acceleration of 0.03–0.05 g were performed among Wenchuan excitations to investigate the change of the modal parameters of embankment slopes. The loading method of shaking table test is shown in Table 3. In which, Wenchuan excitation was code-named WCXZ, and white noise was code-named WN. Three groups of shaking table tests shared a same loading method.

Fig. 3
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Time history (a) and Fourier spectrum (b) of compressed Wenchuan earthquake motion in shaking table test

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Table 3 Loading method in shaking table test
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2.3 Instruments and measurements

The sensors including accelerometers and dynamic earth pressure cells were used in shaking table test. The type of accelerometer was CA-YD-189. The measuring range and the frequency range of accelerometer were ±0.5 g and 0.2–1,000 Hz, respectively. Accelerometers were laid in slope surface and mid-section of embankment slope in both horizontal and vertical directions. The mid-section was a vertical plane which was about 0.4 m distance from the backside of model box. The horizontal accelerometers in slope surface and mid-section were upwards numbered as AH1–H4 and AH5–AH7, respectively. The vertical accelerometers in slope surface and mid-section were upwards numbered as AV1–V4 and AV5–AV7, respectively. Additional, one horizontal accelerometer and another vertical accelerometer were set on rock foundation, and they were numbered as AH0 and AV0, respectively, as shown in Fig. 1.

The type of dynamic earth pressure cell used in the test was BX-7 with a measuring range of 0.1–100 kPa. Three dynamic earth pressure cells were laid in mid-section, and they were upwards numbered as D3, D4 and D5. Another two dynamic earth pressure cells were laid in a horizontal plane which was 0.2 m elevation to rock foundation, and they were numbered as D1 and D2, as shown in Fig. 1.

Both acceleration response and dynamic earth pressure response were collected by Dewetron 2010 Data Acquisition System, and the sampling frequency was 2,000 Hz.

3 Modal parameters of embankment slope

The modal parameters of embankment slope mainly include natural frequency and damping ratio. The change of modal parameters can sensitively reflect the effect of seismic excitation on embankment slope. In this work, the modal parameters of embankment slope were determined by Transfer Function Method based on the acceleration response in slope surface under white noise excitations. Transfer Function Method is a widely used method in Frequency Domain Identification Analysis. The transfer function H a (iω) can be determined as following for shaking table test:

$$H_{a} (i\omega ) = \frac{{S_{xy} (i\omega )}}{{S_{x} (\omega )}}$$
(1)

In which, S xy (iω) is the cross spectral density function between the acceleration response of test point and the acceleration response on rock foundation; S x (ω) is the self-power spectral density function of acceleration response. The natural frequency and damping ratio can be determined by amplitude analysis method based on the results of transfer function H a (iω).

Figures 4 and 5 show the change of modal parameters in both horizontal and vertical directions during the test. The natural frequency of reinforced embankment slope is larger than that of unreinforced embankment slope, and the natural frequency of 4-layer reinforced embankment slope is also larger than that of 2-layer reinforced embankment slope. Geogrids in embankment slope can increase the natural frequency of embankment slope and reduce its damping ratio. The interface friction function between geogrid and soil can restrict the deformation of soil and improve the shear modulus of embankment slope. Thus, the seismic stability of reinforced embankment slope can be improved.

Fig. 4
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First-order natural frequency (a) and damping ratio (b) of embankment slopes in horizontal direction

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Fig. 5
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First-order natural frequency (a) and damping ratio (b) of embankment slopes in vertical direction

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No matter whether the embankment slope is reinforced or not, the natural frequency of embankment slope in both horizontal and vertical directions presents a decreasing trend during the test. Contrarily, the damping ratio exhibits an increasing trend although the test results of damping ratio are somewhat scattered. Such phenomenon reflects that the shear modulus of the embankment slope decreases, and the soil is somewhat loosened during the test. With the increase of excitations, the decreasing ratio of the natural frequency of unreinforced embankment slope is larger than that of reinforced embankment slope. For example, the initial natural frequency of unreinforced embankment slope is 24.0 Hz under WN1 condition. After the test, the natural frequency of unreinforced embankment slope is 18.8 Hz under WN9 condition with a decreasing ratio of 21.7 %. The natural frequency of 4-layer reinforced embankment slope decreases from 25.7 to 22.2 Hz with the decreasing ratio of 13.6 % during the test. It is seen that the reinforced embankment slope is less sensitive to seismic excitation than the unreinforced embankment slope.

4 Horizontal acceleration response

Acceleration magnification is widely used to evaluate the acceleration response of embankment slope. The pseudo-static method for seismic design in current codes is developed based on the information of acceleration magnification. In this work, acceleration magnification is defined as the ratio of the peak value of the acceleration response in embankment slope to the peak value of the acceleration on rock foundation. The distributions of horizontal acceleration magnification are shown in Fig. 6. The horizontal acceleration magnification is always larger than 1.0, and the embankment slope presents obvious amplification effect on input excitations. The horizontal acceleration magnification increases along the height of embankment slope, and the increasing ratio increases at the top of embankment slope, especially when the horizontal acceleration of input excitation is larger than 0.8 g. It is inferred that reinforcing effect at the top of embankment slope should be enhanced to restrict the deformation of soil, especially in the areas of high seismic intensity. However, the difference of horizontal acceleration magnification distribution is not obvious among the different patterns of reinforced embankment slopes.

Fig. 6
figure 6

Distributions of horizontal acceleration magnification under Wenchuan excitations of different intensities. a WCXZ2 (A xmax = 0.2 g). b WCXZ3 (A xmax = 0.4 g). c WCXZ5 (A xmax = 0.8 g). d WCXZ6 (A xmax = 1.0 g)

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Figure 7 shows the relationship between the horizontal acceleration magnification and the input horizontal peak acceleration for different test points. Horizontal acceleration magnification differs under the Wenchuan excitations of different intensities. Generally, the horizontal acceleration magnification presents a decreasing trend with the increase of horizontal peak acceleration. According to the research of Lin and Yang (2013), logarithmic function is recommended to describe such relationship:

$${\text{RATIO}} = a + b\text{ln}(A_{x\hbox{max} } )$$
(2)

In which, RATIO represents the horizontal acceleration magnification; A xmax represents the peak value of input horizontal acceleration; a and b are fitting parameters. The fitting results for several test points are shown in Table 4. It is seen that the fitting parameter b is always negative, which reflects the decreasing trend of RATIO with the increase of A xmax. The absolute value of b reflects the decreasing speed of RATIO. It is seen that RATIO of reinforced embankment slope (both 2-layer and 4-layer reinforced embankment slope) decreases more quickly than that of unreinforced embankment slope with the increase of A xmax, which means that the reinforced embankment slope will present a weaker ground motion than unreinforced embankment slope under an equal strong seismic excitation. The result is also validated in Fig. 8, which shows the typical acceleration time history of point AH4 for both 4-layer reinforced embankment slope and unreinforced embankment slope under WCXZ6 condition (A xmax = 1.0 g). The maximum horizontal acceleration response of 4-layer reinforced embankment slope is about 1.468 g, which is about 51.7 % of that of unreinforced embankment slope. The interface friction function between geogrid and soil can reduce the acceleration response of embankment slope, especially when the seismic excitation is strong. Consequently, it is an effective way to improve the seismic stability of embankment slope by taking reinforcing measure.

Fig. 7
figure 7

Relationship between horizontal acceleration magnification and input horizontal peak acceleration for typical test points. a Point AH2. b Point AH3. c Point AH6. d Point AH7

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Table 4 Fitting results for the relationship between horizontal acceleration magnification and the peak value of input horizontal acceleration
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Fig. 8
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Horizontal acceleration time history of point AH4 for unreinforced embankment slope (a) and 4-layer reinforced embankment slope (b) under WCXZ6 condition

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5 Vertical acceleration response

Most of previously published works mainly focus on the research of horizontal acceleration response for the consideration that the horizontal seismic function will induce greater damage to structure than the vertical seismic excitation. However, the vertical inertia force exists with large intensity for many cases, and it is also a main factor that results in the damage of structure especially in meizoseismal areas.

Figure 9 shows the distribution of the vertical acceleration magnification under Wenchuan excitations of different intensities. In which, A zmax refers to the peak value of input vertical acceleration. Like horizontal acceleration magnification, the vertical acceleration magnification increases along the height of embankment slope. However, the value of vertical acceleration magnification is much smaller than that of horizontal acceleration magnification, and the vertical acceleration magnification is even less than 1.0 at the bottom of embankment slope for some excitation conditions. For example, the range of vertical acceleration magnification of 2-layer reinforced embankment slope is 0.86–1.80 under WCXZ3 condition, while the range of horizontal acceleration magnification is 1.06–2.33. Such phenomenon can be well explained in frequency domain. By making Fourier transform on compressed Wenchuan earthquake motion, it is seen that the significant frequency band of compressed Wenchuan earthquake motion is mainly concentrated at 10–30 Hz, as shown in Fig. 3. According to the test results of modal parameters, the range of the horizontal natural frequency of 2-layer reinforced embankment slope is 22.0–24.5 Hz, and the range of the vertical natural frequency is 35.8–39.2 Hz, as shown in Figs. 4 and 5. The significant frequency of compressed Wenchuan earthquake motion well covers the horizontal natural frequency of the embankment slope, which will induce a resonance phenomenon, and subsequently induces a great acceleration magnification effect in horizontal direction.

Fig. 9
figure 9

Distributions of vertical acceleration magnification under Wenchuan excitations of different intensities. a WCXZ2 (A zmax = 0.13 g). b WCXZ3 (A zmax = 0.27 g). c WCXZ5 (A zmax = 0.53 g). d WCXZ6 (A zmax = 0.67 g)

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Figure 10 shows the typical vertical acceleration time histories (i.e., point AH4) for both unreinforced embankment slope and 4-layer reinforced embankment slope under WCXZ6 condition. The peak value of the vertical acceleration response of 4-layer reinforced embankment slope is 1.179 g, which is about 48.1 % of that of unreinforced embankment slope. The geogrid can reduce the magnification effect of vertical acceleration response. When the seismic wave transmits upwards along the embankment slope, the geogrid exists in embankment slope like a barrier, and the interface friction function between geogrid and soil can partly dissipate the seismic energy. Besides, it is also seen that the geogrid performs more obvious reduction effect on vertical acceleration response than that on horizontal acceleration response.

Fig. 10
figure 10

Vertical acceleration time history of point AH4 for unreinforced embankment slope (a) and 4-layer reinforced embankment slope (b) under WCXZ6 condition

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Figure 11 shows the relationship between the vertical acceleration magnification and the peak value of input vertical acceleration for several test points. Unlike horizontal acceleration magnification, vertical acceleration magnification presents an increasing trend with the increase of excitation intensity for most test points. A linear-elastic system with linear damping will present a constant acceleration magnification regardless of the intensity of excitation. It is seen that the soil presents nonlinear characteristic in vertical direction. However, the acceleration magnification in vertical direction does not change as greatly as that in horizontal direction with the increase of input acceleration. The nonlinear characteristic in vertical direction is not as obvious as that in horizontal direction.

Fig. 11
figure 11

Relationship between vertical acceleration magnification and the peak value of input vertical acceleration for typical test points. a Point AV2. b Point AV3. c Point AV6. d Point AV7

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6 Dynamic earth pressure response

The representative study is made on the dynamic earth pressure response of several test points in 4-layer reinforced embankment slope under WCXZ4 condition, as shown in Fig. 12. The initial earth pressure caused by self-weight of soil is removed to analyze only the dynamic component of earth pressure. The baseline of dynamic earth pressure history fluctuates twice under Wenchuan excitation, which induces the residual earth pressure in embankment slope. The fluctuation of point D5 is most obvious and the vertical earth pressure in this zone is enhanced. The residual earth pressure will probably induce a new deformation in embankment slope. Consequently, the stiffness of the upper part of embankment slope should be enhanced, and the compaction degree of the soil should be highly required. Additionally, it is seen that the first fluctuation of the baseline of dynamic earth pressure response is greater than the second fluctuation. It is inferred that the residual earth pressure will be induced by strong ground excitation, and the value of residual earth pressure will be smaller when the embankment slope experiences another equal excitation.

Fig. 12
figure 12

Dynamic earth pressure response of the typical test points in 4-layer reinforced embankment slope under WCXZ4 condition. a Point D1. b Point D3. c Point D4. d Point D5

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Figure 13 compares the dynamic earth pressure response of point D5 in both unreinforced embankment slope and 2-layer reinforced embankment slope. Like point D5 in 4-layer reinforced embankment slope (as shown in Fig. 12d), the baseline of dynamic earth pressure response fluctuates obviously, and the vertical static earth pressure is enhanced. Meanwhile, it is seen that the intensity of dynamic earth pressure response of reinforced embankment slope is much weaker than that of unreinforced embankment slope. The peak values of dynamic earth pressure response of unreinforced embankment slope, 2-layer reinforced embankment slope and 4-layer reinforced embankment slope are 8.09, 5.05 and 4.99 kPa, respectively, and the ratio is about 1.6:1.0:1.0.

Fig. 13
figure 13

Dynamic earth pressure response of Point D5. a Unreinforced embankment slope. b 2-Layer reinforced embankment slope

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7 Conclusions

The natural frequency of embankment slope presents a decreasing trend during the test, and the damping ratio exhibits an increasing trend. The natural frequency of reinforced embankment slope is larger than that of unreinforced embankment slope under the same seismic condition. The reinforced embankment slope is less sensitive to seismic excitation than unreinforced embankment slope.

Horizontal acceleration response is obviously amplified by embankment slope. The horizontal acceleration magnification increases along the height of embankment slope with a higher increasing ratio at the top of embankment slope. Horizontal acceleration magnification presents a decreasing trend with the increase of excitation intensity, and the decreasing ratio of reinforced embankment slope is higher than that of unreinforced embankment slope.

Vertical acceleration magnification is much smaller than horizontal acceleration magnification because the significant frequency band of compressed Wenchuan earthquake motion does not well cover the vertical nature frequency of embankment slope. The vertical acceleration magnification of reinforced embankment slope is much smaller than that of unreinforced embankment slope, especially under the strong ground motion. The nonlinear characteristic of embankment slope in vertical direction is not as obvious as that in horizontal direction.

Residual earth pressure induced by seismic excitation mainly occurs at the upper part of the embankment slope. The intensity of dynamic earth pressure response of the reinforced embankment slope is weaker than that of the unreinforced embankment slope.