When analyzing the dynamic response features of slopes in time-domain, it can be attained that their acceleration magnification effects have obvious nonlinear changes. The dynamic response characteristics of complex slopes cannot be completely reflected on account to the randomness of waves and the complexity of rock mass. Modal analysis, Fourier spectrum features and peak Fourier spectrum amplitude (PFSA) were used to analyze the response characteristics of layered slope from the perspective of frequency domain, in view of the shortcomings of time domain analysis.

## 4.1 Modal analysis

Modal analysis is an important method of structural dynamic analysis in frequency domain, whose basic governing equation is as follows (Lee and Lee 2012):

$$\left[\text{M}\right]\left\{\ddot{\text{U}}\right\}\text{+}\left[\text{C}\right]\left\{\dot{\text{U}}\right\}\text{+}\left[\text{K}\right]\left\{\text{U}\right\}\text{=}\left[\text{F}\right]$$

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Where, [M], [C], [K] and [F] are mass matrix, damping matrix, stiffness matrix and external load function, respectively. \(\left\{\ddot{\text{U}}\right\}\), \(\left\{\dot{\text{U}}\right\}\)and \(\left\{\text{U}\right\}\) are acceleration, velocity and displacement vectors, respectively. In ideal circumstances, the original equation is changed into the following equation:

$$\left[\text{M}\right]\left\{\ddot{\text{U}}\right\}\text{+}\left[\text{K}\right]\left\{\text{U}\right\}\text{=0}$$

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Hence, the eigenvalues *ω**i* can be obtained by solving the homogeneous differential equation. In accordance with the formula of *f**i* = *ω**i*/2π can get a certain order inherent frequency, where *f**i* for a certain order inherent frequency. Since the dynamic characteristics of model are mainly controlled by low-order modes, the first several modes are mainly considered in modal analysis.

FEM calculation uses linear perturbation analysis step to solve the slope vibration mode, and the modal analysis results of four slopes are presented in Figs. 15–18. Figure 15 shows that the first three vibration modes of Model 1 are bending, shear, and torsion modes, respectively. From the first and second order modes, Model 1 is mainly characterized by overall bending and torsional deformation, and the U (relative displacement) generally occurs at the top slope, which indicates that the dynamic deformation at the slope crest is the most unstable area. Besides, Model 1 presents local torsional and bending deformation in the third mode. The deformation location generally appears in the area of the surface, proving that the slope surface is the most vulnerable area. Figure 16 shows that the first three vibration modes of Model 2 are mainly shear and bending modes. The first mode is mainly including shear and bending deformation, and the Umax appear at the top slope. The second and third modes are mainly large deformation in local scope, and this area mainly appears at the slope crest and junction of structural planes. The modal analysis results of Model 3 show that the U increases with the elevation, and Umax is obtained at the top slope (Fig. 17). The first three vibration modes of bedding slope include bending, shear and torsion modes, and the first mode plays an integral role in controlling the area above the structural planes. The analysis of second and third order modes shows that torsional and bending deformation in the local area above the structural planes can be found.

Figure 18 shows that the first three vibration modes of Model 4 are mainly bending modes. The first mode indicates that the upper part of the slope is subjected to bending deformation, while the second and third modes indicate local deformation of surrounding rock near the structural planes. Figures 15–18 show that with the increase of the modal, the natural frequency of slope gradually, and the deformation characteristics become more complex. By compared with layered slopes, the vibration modes of Model 1 are relatively simple. In low-order modes, the overall shear deformation of slopes is the principal mode. In high-order modes, the local bending and torsional deformation of slopes crest and surface is the main deformation model. Nevertheless, the vibration mode analysis of Model 2–4 show that when shear failure occurs, the slope above the top discontinuity produces a large shear displacement and gradually extends to the adjacent structural plane. Particularly, the various structural planes produce different deformation characteristics. The shear deformation is easy to occur at the top slope in the lower modes. In high-order modes, Model 2 is mainly the shear, torsion, and bedding deformations between each block, Model 3 is the local deformation at the top slope, and Model 4 is the deformation at the middle and upper slope. Besides, the various modes of layered slope show that structural planes amplifies the deformation of the slope. Deformation mechanism of various slopes can also be predicted via modal analysis. The high-order frequency destroys the slope locally and induces the uneven deformation between the slope blocks, forming part of the independent individuals. Low-order frequency controls the overall deformation of slope. Therefore, if subject to continuous earthquakes, the slope damage caused by high-frequency component gradually accumulates, and when it reaches a certain degree, the slope slips and becomes unstable which the holistic deformation is controlled by the lower order mode.

## 4.2 Fourier spectrum analysis

Modal analysis shows that different frequency sections of waves have different influences on dynamic response characteristics of slopes (Fan et al. 2017). To analyze the influences of different frequency segments on dynamic response features of slopes, acceleration time-histories of typical measurement points in Fig. 9 are selected to obtain corresponding Fourier spectrum via FFT (Fig. 19). With the increase of *h*/*H*, the spectral components of slopes become more complex. The Fourier spectrum of the initial input wave is single-peak and has a distinct amplitude curve after 20 Hz. But the Fourier spectrum of response waves are multi-peak, and the Fourier spectral amplitudes of slopes gradually weaken or even disappear with the increase of elevation when the frequency is > 20 Hz, indicating that the slope plays a role of high-frequency filtering on waves.

The first peak ridge of Fourier spectrum of Model 1 generally occurs around 3–5 Hz, and the PFSA increases gradually with elevation (Fig. 19a). Figures 19b-d show that as the elevation of Model 2–4 increases, the occurrence location of the first peak ridge gradually moves to the lower frequency direction. The first dominant frequency *f*1 of Model 2–3 appears between 5–10 Hz, and PFSA increases gradually with elevation, reaching the maximum value at the top slope. In Model 4, range of *f*1 is also 5–10 Hz, but the PFSAmax appears near A13, implying that anti-dip slope is when *h*/*H* is < 0.7, PFSA increased with elevation, PFSA gradually began to decrease when *h*/*H* > 0.7. Therefore, the initial frequency of layered slopes is mainly concentrated in 5–10 Hz, and with the increase of *h*/*H*, the frequency of slopes gradually moves to the left. On the other hand, the Fourier spectral amplitude in the *f*2 is small under the initial stage, but gradually increases with elevation and fluctuates within a certain range. This is because structural planes and the rock near the slope crest easily happen a complex reflection and refraction on the transmission of waves, which leads to the redistribution of wave energy (Cao et al. 2019).

To further analyze the dynamic response characteristics of granite slopes due to topographic and geological effects, MATLAB software was used to perform FFT on the time-history curves of the points shown in Fig. 4 in batches, and the PFSA of *f*1 was extracted. 3D wireframe graphics are present in Fig. 20. Model 1 has an obvious linear increasing trend according to Fig. 20, and PFSA increases gradually with elevation and reaches its maximum value at the top slope. In Model 1, frequency domain (PFSA analysis) and time domain (PGA analysis) have the similar change trend. But the slope change trend is more regular under frequency analysis (Feng et al. 2020), and PGA may have data fluctuation due to the selection of monitoring point location. Model 2 has two obvious bulges in Fig. 20, which is consistent with the analysis results in Figs. 10 and 13, suggesting that it tends to produce energy concentration in the middle and lower part of slope surface and slope crest. PFSA in Model 3 has remarked elevation amplification effect, which increases gradually with *h*/*H* by compared with the PGA. Model 4 has a typical peak value at the *h*/*H* of 0.7, betokening that the middle-upper slope surface area is the most prone to deformation, and attention should be paid to protection in practical engineering. The PFSA has a more obvious bulge where the acceleration increases or decreases sharply, making it easier to identify the slope instability area.

To better investigate the dynamic response and holistic deformation characteristics of slope, Kriging interpolation method was used to plot PFSA contour maps of the four types of slopes are depicted in Fig. 21. The PFSA of all types of slope surface is significantly larger than that of the inner slope, showing that obvious slope surface amplification effect. The maximum values of both Model 1 and 3 are obtained at the slop crest, implying that the mechanical properties of these two models are the most unstable under earthquake. Besides, the PFSAmax of the Model 4 appears near *h*/*H* of 0.7, indicating that this area is a potential failure section subject to earthquake. The Model 2 shows obvious energy concentration at the middle-lower part of slope surface and slope crest, pointing that the Model 2 is prone to failure at these two places under horizontal wave, and may produce penetration failure during continuous earthquake.

Additionally, according to modal analysis and relevant literature investigation (Li et al. 2019b; Song et al. 2019), waves in different natural frequencies have different influences on dynamic response and deformation characteristics of slopes. In this work, Model 1 and 3 are taken as comparative examples, and PFSA cloud maps of *f*2 are drawn to explore the influence of different frequency segments on slope dynamic response, as presented in Fig. 22. PFSA distribution characteristics *f*1 and *f*2 of Model 1 are generally similar, it was seen that some partial amplification area in the waist and inside slope (Fig. 22a) in *f*2 segment, besides the obvious amplification effect at the slope crest. However, the distribution characteristics of PFSA in the two frequency bands of Model 3 are quite different through the comparison results in Figs. 21c and 22b. Model 3 not only has amplification effect at the topmost slope body, but also has amplification effect on the slope surface and other positions inside slope, which is different from the PGA analysis. In conclusion, the *f*1 is the main frequency segment, and the distribution characteristics of PFSA are similar to those of PGA in time domain. Figures 21–22 show that the distribution characteristics of PFSA in the high-order frequency band are different from those PGA and low-order frequency band (*f*1). Compared with Figs. 19–22, structural planes can filter the high-frequency wave to a certain extent and amplify the failure of slope (Li et al. 2019b). Additionally, PFSA changes in the frequency domain may be less affected by the location selection of monitoring sites (Feng et al. 2020).

## 4.3 Dynamic deformation mode analysis of slopes

Structural planes have an important influence on the seismic failure evolution of layered slopes (Fan et al. 2019; Feng et al. 2020). In the time domain, seismic wave will produce complex refraction and reflection near structural planes, causing the uneven change of acceleration on both sides of structural planes, and accelerate the instability of layered slopes. In the frequency domain, the distribution characteristics of slope PFSA are consistent with the variation of the PGA. In addition, the high-order modes cause the local failure of the rock slope, and the damage in the rock mass of the slope continues to accumulate under earthquake (Song et al. 2019). Finally, the slope will produce large-scale instability deformation under low-order natural frequencies that it controls the shear and toppling failure of the whole slope. Figures 19–22 show that structural planes play a filtering role in the face of high-frequency waves, which will accelerate the weakening of the structural plane.

Based on the PFSA isopleth maps and considering the analysis results in time and frequency domain, the seismic instability modes of four models were predicted, as depicted in Fig. 22. The maximum value appears at the top of Model 1, and the slope crest will appear slippage failure along the arc surface under earthquake (Fig. 23a) (Bachmann et al. 2006; Yang et al. 2012). According to DIOR coefficient (the distance between the innermost and outermost peak ridges in FFT spectrum) (Cao et al. 2019) and Fig. 21, the sliding surface of slope instability is multiple composite sliding surface. Model 2 in the slope crest and the middle-lower part of slope surface exist great value, showing that horizontal shear dislocation occurs in the middle and lower part of slope and at the top slope, and then vertical tensile cracks occur in the rock. Finally, the horizontal structural plane intersects with vertical cracks, resulting in step-like failure (Fig. 23b) (Liu et al. 2019). The Model 3 also achieves the maximum value at the slope crest, and its failure mechanism is as follows: first, cracks appear on the topmost structural plane, which promotes the formation of sliding surface; then shear failure occurs along the topmost structural plane of the slope. As the earthquake continues, the sliding body gradually extended downward, and the structural plane of downhill body continued to form a sliding plane, prompting the sliding body to slip on a larger scale (Fig. 23c) (Dong et al. 2022a).The PFSAmax of Model 4 is not at the top slope, but near the *h*/*H* of 0.7, indicating that the Model 4 will produce bending deformation failure in the area, and its failure process is as follows: At first, a large displacement occurs at a relatively high position, leading to the slope foot by upper rock mass extrusion, which gradually produces upward tensile cracks. With the increase of seismic force, through-through cracks eventually form, and toppling failure occurs in the upper rock mass of slope (Fig. 23d) (Ning et al. 2019).

## 4.4 Discussion

Dynamic response characteristics of granite slopes is a multidisciplinary problem, so that the study of rock slope dynamic response based on time domain and frequency domain is helpful to reveal the seismic response characteristics and dynamic failure evolution mechanism of complex rock slope from multiple perspectives. Parameters in time domain analysis have the advantages of easy collection and analysis, such as acceleration time history curve, PGA, etc., can be obtained by sensors. The influence of ground motion parameters and topographic and geological conditions on dynamic response characteristics of slope can also be explored in time domain. Meanwhile, frequency domain analysis can reveal the relationship between wave frequency and dynamic response characteristics of slopes, and can be further verified and supplemented with time domain analysis results. Additionally, previous studies in the frequency domain did not fully consider the internal relationship between modal analysis, seismic frequency band and slope dynamic deformation. In this work, the research content is extended to wave propagation characteristics, spectrum characteristics and slope modal analysis, and the research results in time and frequency domains are fully considered, which further improves the research content in frequency domain. Besides, the dynamic response characteristics and instability processes of various layered slopes are systematically analyzed, the influence of weak structural planes on wave propagation characteristics and the influence of seismic wave frequency on slope dynamic response are revealed, especially from the perspectives of wave propagation characteristics, modal analysis and Fourier spectrum.

Nevertheless, there are some limitations to the work, which this work is only a case study and does not fully consider the dynamic response characteristics of rock slope under complex geological conditions. This work only predicts the slope deformation form based on the time-frequency domain coupled analysis method, which cannot systematically describe the whole process of slope instability and failure, and lacks the analysis and verification of large shaking table or discrete element software. The damage degree of rock under high-frequency sections of wave and the transfer of seismic wave energy at different positions of slope should be considered in the follow-up research.