Introduction

In plant ecological slope protection technology, plant root systems have shallow root reinforcement and deep root anchoring effects, which can effectively enhance the stability of the slope1,2, and have a good reinforcement effect on shallow landslides (sliding surface depth of 1–2 m). In particular, they can reduce soil moisture content and pore water pressure through plant absorption and transpiration, thereby increasing soil shear strength. Moreover, this technology is highly sustainable. The longer the time is, the more favorable it is to control the shallow sliding and collapse of the slope. It is an environmentally friendly protection technology with “vitality”3. Because the root systems of most slope protection plants are relatively shallow, the current plant slope protection technologies are mainly used for shallow landslide prevention4. Therefore, the research team proposed the living stumps slope support structure, which involves planting vigorous seedlings (such as willows or elms) into deep soil layers or inserting their branches into deep soil layers. Over time, the roots grow outward along the woody stems, forming a root-soil composite in the deep soil layer, thus playing a role in slope reinforcement, as shown in Fig. 1. Living stumps are an environmentally friendly form of slope support, which has the potential to prevent and control deep landslides. The existing research results5,6,7,8 show that the safety factor of slopes supported by living stumps is 30–50% higher than that of original slopes under static conditions. It is highly effective for supporting slopes with slip surfaces within 5 m depth and has broad engineering application prospects.

Fig. 1
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Living stumps for slope stabilization.

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It is well known that earthquakes are a major trigger for slope instability. The existence of living stumps will change the waveform of site seismic waves and the propagation characteristics within the slope’s geotechnical body, making the seismic interaction mechanism between living stumps and the geotechnical body very complicated. In order to study the seismic dynamic response of living stumps and geotechnical bodies, R. Sonnenberg9 used linden wood to make rigid roots and rubber to make flexible roots, and these model roots were used to reinforce clay slopes. Through a series of centrifuge model tests, they explored the trends of axial strain and bending strain in the model roots under seismic conditions, and the failure mechanism of clay slopes reinforced by model roots was revealed. Liang10 used 3D printing technology to create model roots and conducted centrifuge model tests to study the response of vegetated slopes during earthquakes. It was found that plant roots reduced the settlement of the slope top by 67% during the sliding process under the action of earthquakes. The centrifuge model test system of plant slope has obvious limitations, such as only generating force in one direction, unable to reproduce complex seismic wave morphology, and difficult to capture the transient dynamic response of slope under seismic action. In contrast, the shaking table model test can accurately reproduce the seismic waves and record the transient response of the slope under seismic action through the sensor. Additionally, domestic and foreign scholars often use Fourier transform11 and wavelet transform12 to analyze seismic signals. Fourier transform is suitable for stationary signals, but it can only obtain the frequency ___domain information of seismic signals and cannot provide time ___domain information. Although wavelet transform can obtain time-frequency ___domain information, its analysis effect depends on the selection of wavelet basis functions, making it difficult to accurately analyze seismic signals13. In contrast, Hilbert-Huang transform (HHT) has the advantages of strong adaptability, strong ability to deal with nonlinear and non-stationary signals, and high time-frequency resolution14. In this paper, a large-scale shaking table model test of living stumps slopes was designed and conducted. With the help of Hilbert-Huang transform, the seismic response characteristics of living stumps slopes are studied from the perspective of time-frequency ___domain and energy ___domain, aiming to provide a basis for related research.

Shaking table test design

Experimental equipment system and its main parameters

The shaking table model test was completed on the multifunctional shaking table system at the National Engineering Laboratory for High-Speed Railway Construction Technology of Central South University, as shown in Fig. 2. This shaking table system consists of a 4 m×4 m six-degree-of-freedom fixed platform and three six-degree-of-freedom mobile platforms of the same size. The key performance parameters are shown in Table 1.

Fig. 2
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Shaking table test.

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Table 1 Main parameters of shaking table performance.
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Model design

Similarity relationships

The reliability of shaking table model tests lies in whether the model can accurately reflect the dynamic response of the prototype structure. Therefore, the geometric similarity, kinematic similarity, and material similarity between the prototype and the model15,16 are crucial. The geometric size, acceleration and density similarity ratio are taken as the main control parameters, and the geometric similarity ratio of the model is determined to be 1 : 7, the acceleration similarity ratio is 1 : 1, and the density similarity ratio is 1 : 1. The similarity ratios of the remaining parameters are derived by Meymand’s similarity laws17, as shown in Table 2.

Table 2 Primary similitude coefficients of model.
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Model box design and boundary treatment

The rigid model box was used in the test, and the internal clearance size was 350 cm\(\:\times\:\)150cm\(\:\times\:\)210 cm (length\(\:\times\:\)width\(\:\times\:\)height). In order to reduce the “model box effect”18, the boundaries of the model box were treated accordingly19: (1) A layer of thick fine stones was laid at the bottom, followed by a layer of fine sand to fill the voids. Then the cement mortar was poured on the top to form the friction boundary, and a layer of 30 cm thick composite material was placed on the top to simulate the bedrock. (2) A 5 cm thick polyethylene foam board was pasted on the inner walls of the model box to create a flexible boundary to weaken the reflection of seismic waves by the model box. (3) Polyvinyl chloride film was pasted on the foam board to reduce the friction between the model and the boundary. The boundary treatment is shown in Fig. 3.

Fig. 3
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Boundary processing. (a) Fine stone layer at the bottom of the box. (b) Sliding boundary and flexible boundary.

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Experimental materials

The stability of living stumps slope is closely related to the morphology of the living stumps’ root system. Yen20 and Gray21 classified root system morphology into six categories, among which the “VH” type has a well-developed main root growing vertically downward, with lateral roots concentrated in the upper third of the main root and extending widely. As shown in Fig. 1, for living stumps slope, the length of the main root is a crucial factor affecting the slope’s anti-sliding capacity, while the direction and lateral extension width of the lateral roots are key factors influencing the “anchor support effect” of the lateral roots7. In view of this, the “VH” type elm root system was selected to make a living stump mode in the experiment.

Although the real plant root systems can more accurately reflect their mechanical and hydrological effects, they are not repeatable in model tests. Studies have shown that acrylonitrile-butadiene-styrene (ABS) material can not only effectively simulate the stiffness and strength of roots, but also facilitate the accurate construction of root models in proportion by 3D printing technology10,22,23. The experiment used ABS material and 3D printing technology to create elm stump models at a geometric similarity ratio of 1:7. Epoxy resin was used as an adhesive to attach a layer of quartz sand to the surface of the living stumps model to simulate the real contact surface between roots and soil. In addition, some studies have shown that rectangular distribution is the optimal pattern for enhancing the stability of living stumps slope24. Therefore, in this experiment, the living stumps were arranged in a rectangular grid of 3 rows and 3 columns inside the slope. The center of the living tree pile matrix was located on the central axis of the slope. The horizontal spacing between each living stump was 38 cm, and the vertical spacing was 30 cm. The living stumps model is shown in Fig. 4.

Fig. 4
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3D printed model of a live stump (Unit: cm).

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The soil sample used in the experiment was cohesive soil, and the large stone is filtered out by a sieve with a diameter of 1 cm to simulate the main part of the slope. A composite material with a thickness of 30 cm was used to simulate the bedrock at the bottom of the slope. The composition ratio of the composite material was barite powder: quartz sand: lithium-based lubricant = 10:5:1. The specific material parameters are shown in Table 3. The model box test of the living stumps slope is shown in Fig. 5.

Table 3 Material physical and mechanical parameters.
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Fig. 5
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Living stumps slope test model diagram.

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Sensor arrangement

Acceleration was measured using uniaxial accelerometers model 1221 L-002, with a range of ± 20 m/s² and a sensitivity of 2000mv/g. Displacement was measured using laser displacement sensors model IL1000, with a range of ± 300 mm. The arrangement of the accelerometer and displacement sensor measurement points is shown in Fig. 6. The accelerometer measurement points were numbered A0-A5 (A0 was located on the shaking table surface to verify seismic wave input). The slope surface displacement measurement points coincide with the accelerometer points and were numbered D1-D5. Considering the bidirectional loading of seismic waves, sensors monitoring dynamic responses in both the horizontal (X-axis) and vertical (Z-axis) directions were arranged at each measurement point. All measurement points were arranged along the central axis of the living stumps slope model.

Fig. 6
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Layout of accelerometer and displacement measurement points (Unit: cm).

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Seismic wave input scheme

The Kobe wave and the Darui artificial wave with a time compression ratio of 2.65 were used as input waves, both of which were bidirectional excited (horizontal + vertical), hereinafter referred to as K-XZ and DR-XZ. Figures 7 and 8 show the time history curves and corresponding Fourier spectrum when the horizontal acceleration is 0.4 g after time compression of these two kinds of seismic waves. The test follows the specification25, and the horizontal acceleration peaks are set to be 0.1 g, 0.2 g, 0.4 g and 0.6 g respectively. The corresponding vertical acceleration peaks are two-thirds of the horizontal peak, namely 0.067 g, 0.133 g, 0.267 g and 0.4 g. The input wave loading sequence is step-by-step loading. White noise is used to scan before and after the test and before the input wave of the same excitation intensity is loaded. The specific loading scheme is shown in Table 4.

Fig. 7
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Acceleration time-history curve (0.4 g). (a) K-XZ. (b) DR-XZ.

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Fig. 8
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Fourier spectrum curves (0.4 g). (a) K-XZ. (b) DR-XZ.

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Table 4 Loading sequences of the test.
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Before formally loading each level of seismic waves, a segment of white noise with amplitude of 0.08 g and a duration of no less than 60 s26 (denoted as WN-1 to WN-5) was input to observe the dynamic characteristics of the model in real-time. Figure 9 shows the horizontal acceleration time history curve of the measuring point A5 of the living tree pile slope after the white noise WN-1 and WN-5 are input into the model. From the comparison of Fig. 9 (a) and (b), it can be seen that under the action of white noise with the same excitation intensity, the horizontal acceleration time history curve of the measuring point A5 before and after the test begins to change little. In addition, the data of the measuring point A5 after the action of WN-1 and WN-5 are changed by fast Fourier transform, and the acceleration Fourier spectrum is obtained, as shown in Fig. 10. Figure 10 (a) and (b) are the Fourier spectrum of the measuring point A5 after WN-1 and WN-5, respectively. The comparison shows that the predominant frequency bands of the two Fourier spectra are 0 ~ 30 Hz, and their morphological distribution is similar, with multiple peaks. Therefore, the internal structural damage of the model can be ignored.

Fig. 9
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Presents the time history curve of horizontal acceleration at point A5 after white noise scanning at different stages. (a) WN-1. (b) WN-5.

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Fig. 10
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Fourier spectrum of A5 after white noise scanning at different stages. (a) WN-1. (b) WN-5.

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Analysis of seismic dynamic response of the model

Some scholars27,28 proposed that the data measured in shaking table tests is susceptible to noise and other interferences, resulting in certain errors. Therefore, before analysis, the wavelet transform filter is used as a low-pass filters to process the data. The data collected by various sensors is analyzed to study the peak acceleration, acceleration amplification factor, and displacement of living stumps slopes under different types and intensities of seismic waves.

Peak acceleration analysis

The seismic waves were bidirectionally loaded, causing acceleration in both the horizontal and vertical directions of the living stumps slope. However, the acceleration response in the horizontal direction is significantly higher than that in the vertical direction, so the analysis of the horizontal peak acceleration is focused. The horizontal peak acceleration of each measuring point of the living tree pile slope is shown in Fig. 11. According to Fig. 11 (a), under the 0.1 g excitation intensity of K-XZ, the horizontal peak acceleration values increase from 1.057 at the toe of the slope to 2.875 at the top of the slope. Under the other three excitation intensities, the horizontal peak acceleration also shows a gradually increasing trend from the toe to the top of the slope. In addition, as the excitation intensity increases from 0.1 g to 0.6 g, the horizontal peak acceleration of each measuring point increases significantly. For example, the horizontal peak acceleration of the slope top measuring point increases from 2.875 to 16.008. This indicates that the horizontal peak acceleration of the living stumps slope increases with elevation, and reaches the maximum at the top of the slope; at the same time, the horizontal peak acceleration is significantly positively correlated with the excitation intensity.

As shown in Fig. 11 (a), the horizontal peak acceleration value increases from 3.638 to 11.347 under the action of K-XZ with 0.4 g excitation intensity. Compared with the excitation intensity of 0.1 g, when the excitation intensity is 0.4 g, the horizontal peak acceleration increases more along the elevation. It is found that when the excitation intensity is 0.2 g, the increase of horizontal peak acceleration is similar to that of 0.1 g; the increase under 0.6 g excitation intensity is similar to that under 0.4 g excitation intensity. The above results show that under the action of seismic waves, the increase of horizontal peak acceleration along the elevation is positively correlated with the excitation intensity. It is noteworthy that when the excitation intensity is low (< 0.4 g), the increase of horizontal peak acceleration along the elevation is small, while at higher excitation intensities (≥ 0.4 g), the increase is significantly larger. This indicates that there is a positive correlation between the amplification effect of horizontal peak acceleration along the elevation and the excitation intensity.

It can be seen from Fig. 11 (b) that under the action of DR-XZ, the horizontal peak acceleration of the living stumps slope increases with the increase of the slope height, and is positively correlated with the excitation intensity, reaching the maximum at the top of the slope. Compared with Fig. 11 (a), the trend of peak acceleration under DR-XZ is similar to that under K-XZ, but the horizontal peak acceleration value under DR-XZ is greater than that those under K-XZ. The above results indicate that the horizontal peak acceleration response patterns of the living stumps slope are mainly influenced by elevation and excitation intensity, and has little correlation with the type of seismic wave.

Fig. 11
figure 11

Horizontal Peak Acceleration of a Living Stumps Slope under Different Earthquake Waves and Excitation Intensities. (a) K-XZ. (b) DR-XZ.

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Analysis of acceleration amplification factor

The ratio of the peak acceleration of each measuring point of the living stumps slope to the peak acceleration of the measuring point (A0) in the same direction is defined as the acceleration amplification factor in this direction. Under different types of seismic waves and different excitation intensities, the horizontal acceleration amplification factors of the living stumps slope are shown in Fig. 12. It can be seen from Fig. 12 that with the increase of slope height, the horizontal acceleration amplification coefficient of the living stumps slope shows the characteristics of nonlinear growth, and there is a significant “acceleration amplification effect”, and the peak value appears at the top of the slope. Furthermore, under different excitation intensities of K-XZ and DR-XZ, the variation patterns of the horizontal acceleration amplification factors are consistent, both of which show an upward trend. The horizontal acceleration amplification factors range from 1.075 to 3.440 under K-XZ excitation, and from 1.005 to 3.735 under DR-XZ excitation.

As shown in Fig. 12 (a), under the action of K-XZ with different excitation intensities, the acceleration amplification factor at the top of the slope is the largest when the excitation intensity is 0.4 g, followed by 0.2 g and 0.6 g, and the acceleration amplification factor under the action of 0.1 g is the smallest. This phenomenon is not only observed at the top of the slope, but also at other measurement points on the slope. It is worth noting that similar results are also observed under the action of DR-XZ with different excitation intensities, which indicates that the acceleration amplification effect has little correlation with the type of seismic wave. Based on the above analysis, the reason for this phenomenon may be that when the seismic wave excitation intensity is low (≤ 0.4 g), the soil of the living stumps slope is re-compaction under vibration. Additionally, due to the low excitation intensity, the internal deformation of the slope is minimal, resulting in less dissipation of seismic wave energy. Consequently, the horizontal acceleration amplification factor of the measurement points increases with the increase of excitation intensity, meaning the amplification effect strengthens with the rise in excitation intensity. However, when the intensity of the seismic wave is high (> 0.4 g), the internal deformation of the slope is large, and the deformation gradually recovers. In this process, the energy dissipation of the seismic wave increases significantly. This leads to a decrease in the horizontal acceleration amplification factor of the measurement points, thereby weakening the horizontal acceleration amplification effect.

Fig. 12
figure 12

Horizontal Acceleration Amplification Factor of a Live Tree Stump Slope under Different Earthquake Waves and Excitation Intensities. (a) K-XZ. (b) DR-XZ.

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Displacement analysis

In the process of seismic wave excitation, the living stumps slope generates horizontal dynamic displacement on the slope surface due to seismic dynamic action. The horizontal peak displacement response of the living stumps slope under different excitation intensities of K-XZ and DR-XZ is shown in Fig. 13. As shown in Fig. 13 (a), under the action of K-XZ with an excitation intensity of 0.4 g, the horizontal peak displacement of the living stumps slope increases from 0.351 mm at the toe to 3.449 mm at the top. Under the other three excitation intensities, the horizontal peak displacement also shows a gradually increasing trend from the toe to the top of the slope. This indicates that the horizontal peak displacement of the living stumps slope shows significant elevation amplification effect. And the dynamic response of the living stumps slope gradually increases from bottom to top, reaching the maximum at the top of the slope. In addition, with the increase of excitation intensity, the horizontal peak displacement increases significantly; and the greater the excitation intensity, the greater the increase of the horizontal peak displacement. For example, when the excitation intensity increases from 0.1 g to 0.6 g, the peak displacement at the top measurement point increases from 0.927 mm to 4.942 mm. As shown in Fig. 13(b), under different excitation intensities of DR-XZ, the horizontal peak displacement response patterns of the living stumps slope are similar to those in Fig. 13(a). This shows that the horizontal peak displacement response of the living stumps slope mainly depends on the elevation and the excitation intensity, and is positively correlated, while it is less correlated with the type of seismic wave.

As shown in Fig. 13, regardless of the type of seismic wave, when the excitation intensity is less than 0.4 g, the horizontal peak displacement of the living stumps slope increases slightly along the elevation; whereas, when the excitation intensity is ≥ 0.4 g, the increase is large. Specifically, under the action of 0.1 g DR-XZ, the horizontal peak displacement increases from 0.347 mm to 0.970 mm along the elevation; while under 0.6 g, it jumps from 0.561 mm to 5.468 mm. This shows that regardless of the type of seismic wave, the increase of horizontal peak displacement along the elevation is mainly related to the excitation intensity.

Fig. 13
figure 13

Horizontal Dynamic Peak Displacement of a Live Tree Stump Slope under Different Earthquake Waves and Excitation Intensities. (a) K-XZ. (b) DR-XZ.

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Considering the difference in the increase of seismic wave excitation intensity in the model test and the lag effect of propagation in the model, the incremental displacement29 is adopted, and it is normalized by the slope height, as shown in Fig. 14. It can be seen from Fig. 14 that no matter what kind of seismic wave, when the excitation intensity is 0.6, the incremental displacement of each measuring point is greater than the value under the other three excitation intensities. Further analysis of Fig. 14, it can be found that no matter what kind of seismic wave action, the incremental displacement of each measuring point shows D2 > D3 > D1 > D4 > D5. It is noteworthy that although the horizontal peak displacement of D5 in Fig. 11 is significantly higher than that of the other four measurement points, the residual displacement caused by the vibration is larger, resulting in the smallest incremental displacement. In contrast, D1 and D2 located at the toe of the slope have a large incremental displacement due to the small residual displacement after the end of the vibration, which indicates that the toe of the slope has good stability and self-healing ability, and can recover most of the peak displacement.

Fig. 14
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Incremental Displacement of a Live Tree Stump Slope under Different Earthquake Waves and Excitation Intensities. (a) K-XZ. (b) DR-XZ.

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Time-frequency analysis based on HHT transformation

The Hilbert-Huang Transform30,31 (hereinafter referred to as HHT) is an advanced tool for analyzing nonlinear and non-stationary time series signals. HHT can effectively capture the spectral characteristics of slopes under seismic excitation, so it is often used as a vibration signal analysis tool. Compared with Fourier transform and wavelet transform, HHT has stronger adaptability, higher time-frequency resolution and complete data-driven characteristics when dealing with nonlinear and non-stationary signals. At the same time, it can reduce the error of Fourier transform and effectively avoid the time-frequency relationship in Heisenberg uncertainty principle. Therefore, HHT has a high adaptability to sudden signals such as earthquakes. By applying HHT to the time acceleration data of the living stumps slope, the dynamic response of the living stumps slope can be analyzed in both the time and frequency domains simultaneously. Additionally, the Hilbert marginal spectrum can be obtained by integrating the HHT spectrum over time. The area under the marginal spectrum represents the total seismic Hilbert energy in the total frequency range, and provides the total seismic Hilbert energy contribution scale of each frequency value. Therefore, the marginal spectrum change can be obtained by the acceleration of each measuring point, and the energy change law of different positions in the process of seismic excitation can be compared.

Hilbert spectrum analysis

The Hilbert-Huang transform (HHT) is used to process the acceleration data of each measuring point at the foot, middle and top of the slope of the living stumps slope under the action of two kinds of seismic waves with different excitation intensities, and the corresponding HHT spectrum is obtained. Figures 15 and 16 show the HHT spectra under the action of K-XZ and DR-XZ when the excitation intensity is 0.2 g and 0.4 g. It can be seen from the figure that the main frequency of the HHT spectrum of the living tree pile slope is distributed between 5 and 15 Hz regardless of the type of seismic wave. Figures 15 and 16 show that regardless of the type of seismic wave, the low-frequency component changes to the high-frequency component during the propagation of the seismic wave to the top of the slope, and the seismic Hilbert energy gradually converges at the top of the slope. This indicates that the living stumps slope has the characteristic of filtering the low-frequency components and amplifying the high-frequency components, and this behavior is less correlated with the type of seismic wave.

Figure 15 (b) shows the HHT spectrum of the living stumps slope under the action of K-XZ with the excitation intensity of 0.4 g, where the PSHEA at the measurement points from the toe to the top of the slope are 2.9, 8.3, and 12.1, respectively. Figure 15 (a) also reveals the gradual increase of PSHEA from bottom to top. This indicates that the PSHEA of the living stumps slope increases continuously along the elevation, and this increasing trend may be related to the filtering of low-frequency components and the amplification of high-frequency components of the living tree pile slope. Comparing Fig. 15 (a) and Fig. 15 (b), when the excitation intensity increases from 0.2 g to 0.4 g, the PSHEA at the top of the slope increases from 6.3 to 12.1. At the same time, the PSHEA at the toe and middle of the slope also increases. This result strongly proves that there is a significant positive correlation between PSHEA and excitation intensity.

Fig. 15
figure 15

HHT spectrum of live tree stump slopes under K-XZ effects at different excitation intensities. (a) 0.2 g. (b) 0.4 g.

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Figure 16 shows the HHT spectrum of the living stumps slope under the action of DR-XZ with excitation intensity of 0.2 g and 0.4 g. The analysis results show that the change trend of PSHEA under the action of DR-XZ with elevation and excitation intensity is consistent with that under the action of K-XZ, that is, it is positively correlated with elevation, but there are numerical differences between the two. This shows that PSHEA is mainly affected by elevation and excitation intensity, and has little correlation with seismic wave types.

Fig. 16
figure 16

HHT spectrum of live tree stump slopes under DR-XZ effects at different excitation intensities. (a) 0.2 g. (b) 0.4 g.

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Marginal spectrum analysis

Through the integration of HHT spectrum with time, the marginal spectrum is obtained and analyzed to explore the variation patterns of seismic Hilbert energy in the living stumps slope. Due to the limitation of space, this paper only presents the marginal spectrum of the slope toe, middle and top of the living stumps slope under the action of K-XZ and DR-XZ with excitation intensity of 0.2 g and 0.4 g (see Figs. 17 and 18).

Figure 17 is the marginal spectrum of K-XZ with excitation intensities of 0.2 g and 0.4 g. According to Fig. 17 (a), when the excitation intensity is 0.2 g, the marginal spectral peak (PMSA) at the toe of the slope is mainly concentrated in the range of 0 ~ 5 Hz.As the elevation increases, the frequency corresponding to PMSA migrates to the vicinity of 10 Hz. Meanwhile, with the increase of slope height, the PMSA from slope toe to slope top is 0.139, 0.574 and 0.814 respectively. In Fig. 17 (b), the value of PMSA also increases from the foot of the slope to the top of the slope. In addition, when the excitation intensity increases from 0.2 g to 0.4 g, the PMSA of the living stumps slope increases significantly, for example, the PMSA at the top of the slope increases from 0.814 to 1.399. This indicates that the PMSA of the living stumps slope is positively correlated with the elevation and increases with the increase of the excitation intensity.

Figure 18 shows the marginal spectrum under the action of DR-XZ with excitation intensity of 0.2 g and 0.4 g. By comparing with Fig. 17, it can be found that the PMSA variation patterns of the living stumps slope under the action of DR-XZ are consistent with that under the action of K-XZ. This indicates that the variation pattern of PMSA is less correlated with the type of seismic wave, which is mainly determined by elevation and excitation intensity. It is noteworthy that the area under the marginal spectrum reflects the total seismic Hilbert energy over the entire frequency range32. As shown in Figs. 17 and 18, with the increase of slope height and excitation intensity, the seismic Hilbert energy gradually accumulates and reaches the maximum at the top of the slope. At the same time, the increase of elevation significantly enhances the filtering effect of the living stumps slope on the low-frequency component of the earthquake and the amplification effect of the high-frequency component, so that the seismic Hilbert energy is transferred from low frequency to high frequency with elevation, resulting in the most significant dynamic response of the slope top. Comparing Figs. 17 and 18, the type of seismic wave has a minor impact on the variation trend of seismic Hilbert energy, only altering its numerical magnitude. In addition, the seismic Hilbert energy is mainly affected by elevation and excitation intensity, and with the increase of elevation and excitation intensity, the seismic Hilbert energy also increases.

Fig. 17
figure 17

Marginal Spectrum of Live Stump Slope under K-XZ Effect at Different Excitation Intensities. (a) 0.2 g. (b) 0.4 g.

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Fig. 18
figure 18

Marginal Spectrum of Live Stump Slope under DR-XZ Effect at Different Excitation Intensities. (a) 0.2 g. (b) 0.4 g.

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Discussion

Numerous scholars33,34,35 have revealed a series of general patterns of seismic dynamic response of slopes through large-scale shaking table tests. Specifically, the peak acceleration, its amplification coefficient and the peak displacement are positively correlated with elevation and excitation intensity. The dynamic response law of the living stumps slope observed in this study is consistent with the research conclusions of many scholars, indicating that the existence of the living stumps does not significantly change the dynamic response law of the slope. However, the existing research on slope displacement mostly focuses on peak displacement, ignoring the gradual increase of seismic wave intensity and its hysteresis effect in the model. Therefore, this paper introduces incremental displacement, as shown in Fig. 14. In this paper, the incremental displacement is analyzed, which effectively supplements the research on slope displacement. The incremental displacement and the peak displacement are mutually verified, which further enhances the reliability of the conclusion.

This paper not only explores the seismic dynamic response patterns of the living stumps slope from the time ___domain, but also obtains the HHT spectra and marginal spectra based on Hilbert-Huang Transform (HHT), and studies time-frequency and energy variation characteristics of the living stumps slope. The research verifies the patterns of the dynamic response of the slope surface acceleration, acceleration amplification coefficient and peak displacement of the living stumps slope observed in the paper, and further proves the influence of the seismic Hilbert energy on the dynamic response of the living stumps slope during the propagation process. The results indicate that as the seismic waves propagate along the slope height, the seismic Hilbert energy continuously accumulates and reaches its peak at the top of the slope. At this time, various parameters representing the seismic dynamic response of the living stumps, such as peak acceleration, show a positive correlation with the elevation, which is most significant at the top of the slope. At the same time, with the increase of seismic excitation intensity, the seismic Hilbert energy also increases, and the dynamic response parameters such as peak acceleration, acceleration amplification factor and displacement also increase synchronously in value. This further illustrates that the dynamic response parameters of the living stumps slope are closely related to the seismic Hilbert energy, indicating that dynamic response parameters such as acceleration are a manifestation of the seismic Hilbert energy.

Conclusions

In this paper, a large-scale shaking table test of the living stumps slope with a geometric similarity ratio of 1:7 was designed and completed. The peak acceleration, acceleration amplification factor and displacement response of the living stumps slope under the action of K-XZ and DR-XZ with different excitation intensities were obtained. Based on the Hilbert-Huang transform, the HHT spectrum and marginal spectrum of the living stumps slope were acquired. The time-frequency and energy variation characteristics were studied, as well as the relationships between seismic Hilbert energy and dynamic response parameters such as peak acceleration, acceleration amplification factor, and displacement. The main conclusions are as follows:

(1) Whether under the action of K-XZ or DR-XZ, the peak acceleration and the acceleration amplification coefficient of the living stumps slope are significantly positively correlated with the relative height and excitation intensity. When the excitation intensity is ≤ 0. 4 g, the acceleration amplification coefficient increases significantly with the increase of excitation intensity, and the acceleration amplification effect is significant. When the excitation intensity is >0.4 g, the acceleration amplification coefficient is less than the value when the excitation intensity is low, and the amplification effect is weakened.

(2) Under the action of different seismic waves, the peak displacement of the soil of the living stumps slope is amplified along the elevation and increases with the increase of excitation intensity. When the excitation intensity is < 0.4 g, the growth rate of the peak displacement along the slope height is relatively gentle. However, when the excitation intensity is ≥ 0.4 g, the growth rate significantly accelerates, and the amplification effect is notably enhanced. In addition, the incremental displacement of the living stumps slope follows the pattern D2 > D3 > D1 > D4 > D5. And the peak displacement is the largest at the top of the slope, but the incremental displacement is the smallest. Conversely, the peak displacement is the smallest at the toe of the slope, but the incremental displacement is relatively larger.

(3) During the propagation of seismic waves, the low-frequency component changes to the high-frequency component, and the living stumps slope shows the characteristics of filtering the low-frequency component of seismic wave and amplifying its high-frequency component.

(4) Under the action of K-XZ or DR-XZ, PSHEA and PMSA increase significantly with elevation and excitation intensity, and reach the maximum at the top of the slope. At the same time, the marginal spectrum is further studied and analyzed. The seismic Hilbert energy shifts from low frequency to high frequency with the increase of height, and gradually accumulates with the slope height, reaching the maximum at the top of the slope. The values of the dynamic response parameters of the living stumps slope are closely related to the seismic Hilbert energy, that is, with the upward accumulation of the seismic Hilbert energy, the values of the dynamic response parameters such as the peak acceleration, the acceleration amplification factor and the displacement of the living stumps slope increase significantly.