Numerical investigation on seismic dynamic response characteristics of high-steep layered granite slopes via time-frequency analysis

doi:10.21203/rs.3.rs-1871751/v1

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Numerical investigation on seismic dynamic response characteristics of high-steep layered granite slopes via time-frequency analysis

https://doi.org/10.21203/rs.3.rs-1871751/v1

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published 13 Mar, 2023

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The occurrence of geological structure and stratum lithology plays important roles in the seismic stability of complex slopes, which is also the difficult points of engineering construction. Four three-dimensional layered granite slope models with infinite boundary were modeling via finite element method (FEM). Seismic response and deformation characteristics of slopes are systematically studied through time-frequency domain. A frequency-domain analysis method of complex slopes is proposed including modal and spectrum conjoint analysis. Modal analysis can directly display the main vibration modes of slopes. The combination of modal and spectral analysis can clarify the inherent characteristics of slopes, and reveal the interaction mechanism between inherent frequency and dynamic characteristics of slopes. The results illustrate that structural faces have significant effects on the propagation characteristics of wave within rock mass, and complex refraction/reflection phenomena occur near these discontinuities, thus to different dynamic response characteristics in the slope. Layered slopes have apparent magnification effect of slope surface and altitude. Directions of seismic excitation and structural plane types affect the dynamic response of slopes. Horizontal wave mainly affects the middle and upper part of high-steep slope, while vertical wave has obvious influence on the slope crest. Additionally, Fourier spectral analysis shows that structural planes have filtering effects on the high-frequency waves. Combined with modal analysis, it further explains that high-frequency section of waves mainly triggers local deformation of slopes, while low-frequency component controls their overall deformation. Finally, instability regions and evolution process of different layered slopes were predicted based on time-frequency conjoint analysis.

High-steep granite slope

Structural planes

Seismic response characteristics

Deformation mechanism

Time-frequency conjoint analysis

Earthquake-induced landslides have become one of the global natural disasters, which often causes plentiful casualties and property losses (Owen et al. 2008; Wasowski et al. 2011; Gischig et al. 2015; Mineo et al. 2015; Haneberg et al. 2022). On 22 June 2022, a 5.9-magnitude earthquake struck eastern Afghanistan, causing massive landslides and killing at least 500 people (USGS 2022). The Chi-chi earthquake in Taiwan (Mw = 7.6) killed 2,321 people and caused 26,000 landslides in September 1999 (Xue 2000; Wasowski et al. 2011) and the Wenchuan earthquake (Mw = 7.9) caused massive slope damage, including 56,000 landslides or rock collapses, and 256 barrier lakes were formed in 2008 (Wasowski et al. 2011; Huang et al. 2013). The complexity of geological structure, lithology, topography and landform are typical characteristics of high-steep slopes in mountainous areas of China, which makes the seismic stability of slopes extremely complicated. The seismic response characteristics and instability mechanism of mountain under specific topographic and geological conditions have been one of the important and difficult problems in engineering geology.

In recent years, many scholars have carried out relevant studies on the dynamic response characteristics and failure modes of rock slopes. Guo et al. (2017) derived the analytical solution of limit equilibrium considering the connection of regional joints in slope bedrock and the transfer force between adjacent blocks. Qi et al. (2015) presented an analytical solution of bedding slope failure considering water pressure and bending deformation based on the energy balance theory. Yagoda-Biran and Hatzor (2013) proposed a three-dimensional instability model diagram of slope toppling and sliding failure by quasi-static analysis. However, theoretical analysis is difficult to solve slope problems with complex contact surfaces and boundary conditions. Model tests and numerical simulation have been the two popular analysis methods. He et al. (2021) selected a high-steep slope in Qinghai-Tibet Plateau as the prototype and studied the dynamic characteristics and failure evolution process of bedding slopes. Huang et al. (2013) explored the similarities and differences in response characteristics of soft and hard rock slopes under earthquakes by using large shaking table. Besides, some scholars also used shaking table tests to explore the dynamic response characteristics of bedding, anti-dip, horizontal and homogeneous slopes, respectively (Liu et al. 2013, 2014; Li et al. 2019c). In numerical simulation, researchers usually use FEM (Pelekis et al. 2017; Li et al. 2019a; Song et al. 2021a), discrete element method (DEM) (Yuan et al. 2015; Gischig et al. 2016; Zheng et al. 2021) and other software to analyze the peak acceleration and failure modes of slopes. Furthermore, previous research schemes mainly focused on the dynamic response characteristics of a single or two types of slopes failure modes, which lacked the systematic analysis of the dynamic response characteristics and wave propagation characteristics of various layered slopes. Meanwhile, as seismic waves are nonlinear and unstable signals, they will have complex interaction with discontinuous and uneven rock mass in the propagation process (Mineo et al. 2015), therefore, it is necessary to systematically analyze the effects of different slope structures on seismic wave propagation characteristics.

Whereas, the above researches mainly focus on the failure evolution process and time-domain parameter analysis of slopes, such as PGA (peak ground acceleration), permanent displacement etc. But as typical non-stationary signals, time and frequency are two basic physical quantities to describe their changes. Seismic waves with different frequencies are significantly correlated with the dynamic response characteristics of layered slopes (Mineo et al. 2015; Li et al. 2019c; Yang et al. 2021; Dong et al. 2022a), and the interaction mechanism between structural planes and wave become complex, thus making it difficult to interpret the dynamic stability. Hence, only studying the variation law of waves in the time-domain can not fully reflect the internal mechanism of failure mode and the dynamic characteristics of slopes.

At present, some scholars began to investigate the dynamic response characteristics of slopes from the perspective of frequency domain or energy dissipation of seismic wave in order to more fully explain the interaction between waves and slopes (Fan et al. 2016, 2017; Song et al. 2020, 2021b). Song et al. (2021c) explored the dynamic response characteristics of slopes under the action of reservoir through frequency and time-frequency domain. Li et al. (2019b) carried out frequency domain analysis on slope with complex geological structure using Fourier spectrum in the way of combining shaking table and DEM, and reveal damage characteristics and failure mode of slopes. Fan et al. (2017, 2019) proposed a calculation method of slope safety factor based on wave energy principle and HHT transformation. In summary, time-frequency analysis can clearly describe the energy density or intensity of seismic waves in different time and frequency ranges, and gradually attracted more and more attention of researchers. However, previous experiments have not conducted in-depth studies on the frequency domain, and model analysis is seldom used to study dynamic characteristics of complex slopes. Besides, Modal analysis is a common method to study the dynamic characteristics of engineering entities in engineering vibration field (Song et al. 2019). Modes refer to the natural vibration characteristics of engineering entities, and each mode has a specific inherent frequency, damping ratio and modal shape. That is the modal analysis which the process of analyzing these modal parameters. FEM is one of the common modal analysis methods. Above all, combining modal analysis with spectrum analysis can further improve the frequency domain analysis framework, which is necessary to interpret the dynamic deformation characteristics of slopes from multiple perspectives.

Layered granite slope is one of the common geologic bodies in nature. Taking layered granite slopes in high-intensity earthquake region of the southeast China as examples, four three-dimensional FEM numerical models are established, including homogeneous slope, horizontal stratified slope, bedding slope and anti-dip slope. A frequency-domain analysis method is proposed including modal and spectrum conjoint analysis. Seismic response characteristics and instability deformation modes of layered slopes are systematically studied from multiple perspectives of time and frequency domains based on the three-dimensional FEM dynamic analysis (Fig. 1). The characteristics of wave propagation in slopes, and the topographic and geological dynamic amplification effect of slopes are discussed by analyzing the acceleration distribution characteristics in time domain. Then, the mechanism of wave propagation characteristics and dynamic amplification effect of different excitation directions on the slopes is compared. Moreover, dynamic amplification effect and deformation characteristics of layered slopes are studied via modal and Fourier spectrum analysis. The interaction between inherent frequency and dynamic deformation characteristics of slopes are clarified, from the perspective of frequency domain. The effects of structural plane types on the dynamic amplification effect and instability mode of slopes are revealed based on the time-frequency coupling analysis. This work can provide reference for seismic stability analysis and seismic reinforcement of similar engineering rock slope.

2.1 Case study

The original slope is located in southeast coast of China, bordering the Taiwan Strait, as shown in Fig. 2. The main terrain around the slope is hilly mountains, valleys crisscross, vegetation development, slope of 50–70°, and rolling stones at the top slope are relatively developed, as presented in Fig. 3a. According to the geological survey, there are many groups of shear joints and cleavages in the rock mass on both sides of the joint dense zone. There are local breccia zones and chlolitization phenomenon of mylonite, and some scratches are found, reflecting the thrust or compressional nature of the main thrust signs. The lithology of slope is medium-grained granite of biotite in late Yanshanian, which is gray and fleshy red, massive structure and medium-grained granitic structure, and the mineral composition includes biotite, quartz, potassium feldspar, plagioclase and so on. According to the weathering degree of the parent rock, it can be divided into strongly weathered, moderately weathered and slightly granite from top to bottom successively. There are multiple weak structural planes in the slope is shown in Fig. 3b. According to the data of the USGS (United States Geological Survey), the largest earthquake occurred near the study area in recent years was M_W=5.7, with a focal depth of 16km (USGS 2022). Other earthquake areas have been marked in Fig. 2. Additionally, this research area is also a site selection area for mountain tunnel engineering in the future, which Stereographic projection of surrounding rock at tunnel mouth is shown in Fig. 3c, hence, it is of engineering significance to study the seismic stability of the slope.

2.2 Numerical modelling

Most of the weak structural planes are distributed in the weathering granite area (Fig. 3b). To study the dynamic response features of high-steep slopes, the original slopes are generalized into four typical models: Model 1 (homogeneous slope), Models 2 (horizontal layered slope), Model 3 (bedding slope), and Model 4 (anti-dip slope), respectively (Fig. 4). The generalized slope Models 1–4 are mainly composed of moderately weathered granite and weak structural planes, and both slopes are 60°. Besides, the angle between the weak structural planes and the horizontal plane in Model 3 are 20°, and that in model 4 are 160°. Physical and mechanical parameters of moderately weathered granite and weak structural plane were measured by a series of rock mechanics experiments in the laboratory including direct shear tests, uniaxial compression tests, and so on (Table 1).

Table 1

Physical and mechanical parameters of rock materials
Material parameter	Density (kg/m³)	Elastic modulus (GPa)	Poisson ratio	Friction angle (°)	Cohesive force (kPa)
Granite	2500	20	0.26	45	1200
Structural planes	2100	0.1	0.32	30	35

FEM was used to explore the dynamic response characteristics of different layered slopes, and four numerical models with infinite element boundaries were established according to the geological profile (Fig. 3b) and generalized geological model (Fig. 4), as exhibited in Fig. 5. In the FEM dynamic analysis, wave transmission and reflection are not only related to the predominate frequency of the input wave, but also closely related to the thickness, length, spacing and size of the weak interlayer (Song et al. 2020; Dong et al. 2022b). Therefore, the models are 900×400×450m (long×wide×high), and the thickness of structural planes is set to 2m, to meet the requirements of dynamic analysis. Rocky slope body adopts 8-node hexahedron element and the structural plane adopts single-layer grid to ensure the calculation accuracy under static and dynamic condition, in which final grid numbers generated by Model 1–4 are 265300, 220400, 271650 and 249550, respectively.

Boundary conditions for FEM simulation solving dynamic problems are generally set as truncated boundary, viscoelastic boundary and infinite element boundary, etc (Çadır et al. 2021; Yang et al. 2021). The truncated boundary can satisfy the general static analysis, but in the dynamic analysis, the truncated boundary has a complete reflection effect on the seismic wave, which leads to the distortion of the results of the measured points in slopes. Viscoelastic boundary is absorbed by applying a sticky pot and a spring in the normal direction and horizontal direction of the boundary respectively. The boundary of infinite element is from finite field in local coordinate system to infinite element in global coordinate system. By comparing various boundary conditions, using the near field district is simulated by FEM and the far field area is simulated by infinite element method, so as to reduce or eliminate the reflection of waves at the boundary surface. The positions of basic nodes and boundary of infinite elements in the four numerical models are shown in Fig. 5, and partial shape functions are shown in Equations (1)-(2) (Bettess 1977; Kumar 2000). Implicit analysis is used in dynamic analysis, and dynamic response characteristics of slope under small deformation condition are only studied. Additionally, the model material is regarded as elastic material in the finite element dynamic analysis, and the dynamic response characteristics in the elastic domain of the slope are taken into account. The layered rock mass material of the slope complies with the Mohr-Coulomb criterion, and its specific physical and mechanical parameters are shown in Table 1.

$${\text{N}}_{\text{i}}^{\text{'}}\text{=0.25}\left(\text{1+η}{\text{η}}_{\text{i}}\right)\left(\text{1+ξ}{\text{ξ}}_{\text{i}}\right){\text{f}}_{\text{1}}$$

$${\text{N}}_{\text{i}}^{\text{'}}\text{=0.25}\left(\text{1+η}{\text{η}}_{\text{i}}\right)\left(\text{1+ξ}{\text{ξ}}_{\text{i}}\right){\text{f}}_{\text{2}}$$

Horizontal and vertical seismic loads were applied to the bottom interface of models to explore the dynamic response features of slopes under different excitation directions. Seismic load is a synthetic wave provided by the local earthquake department, and its acceleration time-history curve and the corresponding Fourier spectrum as depicted in Fig. 6. The duration of wave is 41.6s, the peak acceleration is 0.1g, the preeminent frequency is 3–5 Hz, and the interval time is 0.02s. Additionally, to avoid the influence of gravity, the model first carries out ground stress balance analysis step, before dynamic analysis.

3.1 Characteristics of wave propagation in rock mass

Horizontal and vertical synthetic waves with peak acceleration of 0.1g were respectively loaded at the bottom of the models, and certain a complete propagation process of wave from the bottom to the top of slopes was selected to explore the influence of the structure planes on the propagation characteristics of waves (Figs. 7 and 8). Figure 7 shows Model 1–4 have obvious layered propagation characteristics in the bedrock area, and have obvious elevation and surface amplification effects in the slope body, when input the x direction. Figure 7a shows that the propagation law of horizontal wave in Model 1 is relatively simple, and presents longitudinal propagation and slope surface amplification effect within the slope body. Seismic wave mainly propagates between the blocks formed by the cutting of structural planes, and gradually increases with the increase of elevation in the Model 2–4, that be identified Figs. 7b-d. Besides, compared with Model 2–4, Model 3 (bedding slope) has deeper penetrating area and stronger effect. In Model 4, the horizontal wave propagates gradually upward along the block, but the amplification area is concentrated in the upper part of the slope surface rather than at the top slope, and the local amplification effect is easy to occur at the slope waist.

Figure 8 shows that the models also have typical elevation and surface amplification effect when input wave in the z-direction. The propagation law of wave in Model 1 is simple. The amplitude of waves increases with the slope height, and reaches the maximum at the top slope, and the amplification area mainly concentrates on the slope surface. Wave mainly propagates successively along the structural planes and forms an amplification region on the slope surface for the Model 2–4. Figures 8b-d illustrate that there is obvious local weakening effect and acceleration phase shift between structural planes, indicating that structural planes have a significant effect on the propagation characteristics of waves. Additionally, under vertical wave, the influence range of the middle and lower part of the high-steep slope is small, and the top slope is prone to form instability failure. It can be found that the influence range of horizontal seismic wave is larger than that of vertical wave (Li et al. 2019b). Besides, Vertical waves have obvious irregular propagation characteristics in the slope body, and the local acceleration amplification area mainly concentrates on the edge and the top slopes by compared with the horizontal waves. Figures 7 and 8 show that wave propagation characteristics of Models 2–4 are more complex than that of Model 1. Seismic wave mainly transmits successively along structural planes, and there are obvious acceleration phase shifts and local amplification/weakening phenomena between structural planes. This is because the mechanical properties of structural plane and surrounding rock differ greatly, which leads to complex reflection, refraction and energy redistribution of wave at the interface of discontinuity and slope surface.

3.2 Effect of topography and geology on the dynamic response of slopes

To explore the dynamic response characteristics of slopes under the action of structural planes and earthquake, 21 monitoring points were arranged in the models (Fig. 4). Taking input wave in x-direction as examples, the acceleration time-histories of typical monitoring points at the slope surface were extracted (Fig. 9), showing that the acceleration increases gradually with elevation. In addition, the acceleration waveform of Model 1 is similar to that of input artificial wave, while the waveform of layered slope displays obvious peak differentiation and PGA time delay occurs with the increase of elevation. The change of PGA with h/H is shown in Fig. 10. The relative elevation (h/H) is defined as the ratio of elevation at a point in the slope to the total elevation of slope. PGA curves of Model 2–4 have obvious nonlinear change characteristics, except for the Model 1.

Figure 10 shows that, when input x-direction wave, the Model 2–4 has typical abrupt changes near the h/H of 0.2, 0.4, 0.6 and 0.8, and the elevations are located near structural planes. M_PGA is defined as the ratio of PGA at a certain point of slope to PGA at the foot of slope (Eq. 3), reflecting the acceleration amplification effect at one point of slopes.

$${\text{M}}_{\text{PGA}}\text{=}{\text{PGA}}_{\text{i}}/{\text{PGA}}_{\text{s}}$$

Where ${\text{PGA}}_{\text{i}}$ is PGA at a certain point of slope and ${\text{PGA}}_{\text{s}}$ is PGA at the slope foot. Figure 10a displays that the acceleration amplification interval of Model 2–4 is different at slope surface and inside slope. For example, the M_PGA of the Model 3 is 1.145 when h/H is 0.2–0.4 at the slope surface, while that inside slope increases first and then decreases, attenuating by 21.65%. Under the same condition, the M_PGA of Model 4 slope surface from the foot to the lowest structural plane is 1.836, with an attenuation of 24.67% inside slope. In addition, Fig. 10b shows that the Model 2–4 also have an obvious nonlinearity change trend under vertical wave, and the PGA mutation position is the same as that under horizontal wave. This suggests that structural planes have the effect of enhancing or weakening the seismic amplitude (Cao et al. 2019). Due to the position and inclination angle of structural planes of slopes are different, resulting in superposition or cancellation during the process of waves, and thus to the difference of PGA changes or seismic energy distribution of different layered slopes. In Fig. 10, the acceleration of high-steep layered slope under earthquake does not increase linearly with elevation, but increases and decreases rhythmically (Zhan and Qi 2017), showing an overall increasing trend within the range of slope surface. In particular, PGA change have more complex fluctuation phenomenon with input x-direction wave by comparison with z-direction. Besides, the M_PGAmax of Models 1–4 are 2.124, 4.08, 6.46 and 3.32 when input horizonal wave, respectively. The M_PGA of layered slopes much larger than that of homogeneous slope and bedding slope has the largest M_PGA, suggesting that bedding slope is more prone to instability.

To clarify the influence of structural planes types on the dynamic response characteristics of slopes, the PGA of layered slopes to that of Model 1 (homogeneous slope) is shown in Fig. 11. The ratio of Models 2–3 to Model 1 is > 1.0 overall, indicating that layered slopes have a larger magnification effect. The maximum PGA ratios of Model 2–4 are 1.87, 2.38 and 2.18 (Fig. 11a), respectively, indicating that the dynamic response characteristic of Model 3 is more obvious, and instability is more prone to occur. The similar rule also applies to the input of vertical waves. In particular, M_PGA of slope surface is significantly larger than that of slope interior, presenting a typical surface amplification effect, by comparing the four change curves in Fig. 11. The results bespeak that there are significant differences between the acceleration response of slope surface and inside slope under dynamic action, among which the difference of Model 3 is the largest. The uneven acceleration response further aggravates the instability of the slope surface.

3.3 Influence of input direction of seismic wave

To explore the influence of seismic excitation directions on the dynamic response of slopes, the changes of the ratios of PGA subject to x-direction wave to that subject to z-direction are shown in Fig. 12. In Model 1, the ratios of PGA show a decreasing trend, but the ratios are > 1.0, showing that the horizontal wave affects homogeneous slope greater. In Model 2–3, the ratios of PGA at slope surface fluctuate in the range of 1 to 2.5, and 0.5 to 2 inside slope. In particular, PGA ratios of surface are < 1.0 after h/H = 0.7. The ratios of PGA have typical nonlinear variation characteristics, and as the h/H increases, the ratios of PGA gradually decrease. The results show that the effect of horizontal wave is obvious when the h/H is < 0.7, while the effect of vertical wave is more obvious near the slope shoulder.

Furthermore, inside and outside of Model 2–4 both have extremely uneven acceleration responses under different wave direction, while Model 3 has the most obvious amplification effect. Such as the PGA ratio of slope surface to that inside Model 3 at A21 was calculated, which were 2.8414 and 1.9659 under horizontal and vertical directions, respectively. The results illustrate that, the movement trend of Model 3 is more strongly non-uniform subject to horizontal wave, and prone to generate the relative displacement, further causing instability failure. Hence, Figs. 10–12 points that vertical wave has more influence on layered slope crest, compared with horizontal wave. But the internal and external changes of slope are more nonuniformity under horizontal wave, proving that the inhomogeneity of the internal and external response of slope may also be one of the reasons for its failure and instability, in addition to M_PGA (Zhan and Qi 2017).

3.4 Dynamic stability analysis of slope

PGA of layered slopes have typical nonlinear change characteristics in the time-domain. The Kriging interpolation method is used to draw the PGA distribution contour diagram of slopes (Figs. 13–14), to confirm the peak and failure area of slopes more clearly and intuitively. Figure 13 shows that PGA_max of Model 1 appears at the top slope subject to x-direction wave, proving that slope crest is prone to instability, and the bright yellow part in Fig. 13a constitutes a distinct shear slip zone. There are two obvious acceleration amplification area at the waist and top of Model 2, which indicates that the Model 2 is easy to produce initial fracture failure in this area, and finally the upper and lower regions of slope form a through zone to produce slip deformation. PGA_max of Model 4 does not appear at the top slope, but at the h/H of 0.6–0.7. Figure 13d shows that the motion state of Model 4 near slope crest, interior, and rock mass adjacent the h/H of 0.7 are not synchronized, easily resulting in toppling damage. PGA_max of Model 3 is the largest and appears at the top slope by compared with the other models, which states that the Model 3 is the most unstable in mechanical properties and prone to geological disasters subject to earthquake.

In Fig. 14, the PGA_max of four slopes all appear near the top slope under vertical wave, and the influence range is small. The slope crest is prone to instability failure when the earthquake intensity reaches a certain value. The PGA_max of Model 3 is also the largest in the four models. Besides, the PGA_max of Model 1 subject to horizontal wave is 1.92, which is higher than that subject to vertical, approximately 1.6. the amplification effect of homogeneous slope under horizontal wave is more obvious, and the slope surface area is more prone to slip failure (Figs. 13a and 14a). Figures 13–14 show that the influence range of slopes under horizontal wave are wider, and the damage range of various slopes are larger (Li et al. 2019c), among which Model 3 is the most prone to instability. The PGA_max at the top slope is the largest under vertical wave. These results are consistent with the analysis results in Figs. 10–12.

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):

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

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:

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

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 U_max 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 U_max 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 f₁ 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 f₁ is also 5–10 Hz, but the PFSA_max 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 f₂ 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 f₁ 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 PFSA_max 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 f₂ are drawn to explore the influence of different frequency segments on slope dynamic response, as presented in Fig. 22. PFSA distribution characteristics f₁ and f₂ of Model 1 are generally similar, it was seen that some partial amplification area in the waist and inside slope (Fig. 22a) in f₂ 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 f₁ 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 (f₁). 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 PFSA_max 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.

The time-frequency analysis method is adopted to analyze the dynamic response characteristics of layered granite slopes from multiple perspectives. Some relevant conclusions are as follows:

1. Weak structural planes have a significant influence on the propagation characteristics of waves, and waves will be superimposed or cancelled at the structural plane. PGA curve of layered slopes has obvious nonlinear variation characteristics, and the mutation location is usually near the structural planes. Additionally, bedding slope has the maximum dynamic response or PGA_max, while homogeneous slope has the minimum change. Moreover, structural planes have an important effect on the seismic deformation features and failure mode of slopes, which is the potential slip surface or fissure initiation area. Homogeneous slope generally presents arc landslide. The deformation of horizontal layered slope is mainly caused by the shear dislocation each of structural planes. Structural planes of anti-dip slope easily produce bending deformation under earthquake, which leads to topple failure of slope.

2. The directions of ground motion have effects on the dynamic response characteristics of slopes. PGA ratios of slope surface under horizontal wave to that under vertical wave range from 1-2.5, and inside slope is 0.5-2. Horizontal wave has a great influence on the lower part of layered slopes, while the vertical wave has a great influence on the upper part of slopes. In particular, the uneven response inside and outside the slope is stronger under horizontal wave, such as the PGA ratio of slope surface to inside bedding slope were 2.8414 and 1.9659 under horizontal and vertical directions, respectively. Hence, not only the acceleration amplification effect at different positions, but also the inhomogeneity of acceleration of slope surface and inside slope should be considered, in the time-domain analysis.

3. According to modal and Fourier spectrum analysis, layered slopes have obvious slope surface and structural planes amplification effect. The occurrence of fractures in structural planes and slope crest will lead to complex refraction and reflection in wave transmission process, thus to change of wave components, under continuous earthquake. Structural planes have a certain filtering effect on the high-frequency components of waves (>20 Hz), while the high-frequency component of waves further accelerates the damage of local slope and structural planes, and finally produces the overall deformation induced by low-frequency components of waves.

Acknowledgments

This study was funded by the National Natural Science Foundation of China (52109125, and 41941019), the Open Research Fund of SINOPEC Key Laboratory of Geophysics (WX2021-01-12), the National Key Research and Development Plan (2018YFC1504801), the China Postdoctoral Science Foundation (2020M680583), and the National Postdoctoral Program for Innovative Talent of China (BX20200191), and Graduate Education Reform and Quality Improvement Project of Henan Province (No. YJS2021JD13).

Confict of interest: The authors declare that they have no confict of interest regarding the submitted article.

Bachmann D, Bouissou S, Chemenda A (2006) Influence of large scale topography on gravitational rock mass movements: New insights from physical modeling. Geophys Res Lett 33:1–4. https://doi.org/10.1029/2006GL028028
Bettess P (1977) Infinite elements. Int J Numer Methods Eng 11:53–64. https://doi.org/10.1002/nme.1620110107
Çadır CC, Vekli M, Şahinkaya F (2021) Numerical analysis of a finite slope improved with stone columns under the effect of earthquake force. Springer Netherlands
Cao L, Zhang J, Wang Z, et al (2019) Dynamic response and dynamic failure mode of the slope subjected to earthquake and rainfall. Landslides 16:1467–1482. https://doi.org/10.1007/s10346-019-01179-7
Dong J, Wang C, Huang Z, et al (2022a) Shaking Table Model Test to Determine Dynamic Response Characteristics and Failure Modes of Steep Bedding Rock Slope. Rock Mech Rock Eng. https://doi.org/10.1007/s00603-022-02822-x
Dong L, Song D, Liu G (2022b) Seismic Wave Propagation Characteristics and Their Effects on the Dynamic Response of Layered Rock Sites. Appl Sci 12:1–17. https://doi.org/10.3390/app12020758
Fan G, Zhang J, Wu J, Yan K (2016) Dynamic Response and Dynamic Failure Mode of a Weak Intercalated Rock Slope Using a Shaking Table. Rock Mech Rock Eng 49:3243–3256. https://doi.org/10.1007/s00603-016-0971-7
Fan G, Zhang L, Zhang J, Yang C (2019) Analysis of Seismic Stability of an Obsequent Rock Slope Using Time–Frequency Method. Rock Mech Rock Eng 52:3809–3823. https://doi.org/10.1007/s00603-019-01821-9
Fan G, Zhang LM, Zhang JJ, Yang CW (2017) Time-frequency analysis of instantaneous seismic safety of bedding rock slopes. Soil Dyn Earthq Eng 94:92–101. https://doi.org/10.1016/j.soildyn.2017.01.008
Feng ZY, Huang HY, Chen SC (2020) Analysis of the characteristics of seismic and acoustic signals produced by a dam failure and slope erosion test. Landslides 17:1605–1618. https://doi.org/10.1007/s10346-020-01390-x
Gischig V, Preisig G, Eberhardt E (2016) Numerical investigation of seismically induced rock mass fatigue as a mechanism contributing to the progressive failure of deep-seated landslides. Rock Mech Rock Eng 49:2457–2478. https://doi.org/10.1007/s00603-015-0821-z
Gischig VS, Eberhardt E, Moore JR, Hungr O (2015) On the seismic response of deep-seated rock slope instabilities - Insights from numerical modeling. Eng Geol 193:1–18. https://doi.org/10.1016/j.enggeo.2015.04.003
Guo S, Qi S, Yang G, et al (2017) An analytical solution for block toppling failure of rock slopes during an earthquake. Appl Sci 7:1–17. https://doi.org/10.3390/app7101008
Haneberg WC, Johnson SE, Gurung N (2022) Response of the Laprak, Nepal, landslide to the 2015 Mw 7.8 Gorkha earthquake. Nat Hazards 111:567–584. https://doi.org/10.1007/s11069-021-05067-z
He J, Qi S, Zhan Z, et al (2021) Seismic response characteristics and deformation evolution of the bedding rock slope using a large-scale shaking table. Landslides 18:2835–2853. https://doi.org/10.1007/s10346-021-01682-w
Huang R, Zhao J, Ju N, et al (2013) Analysis of an anti-dip landslide triggered by the 2008 Wenchuan earthquake in China. Nat Hazards 68:1021–1039. https://doi.org/10.1007/s11069-013-0671-5
Kumar P (2000) Infinite elements for numerical analysis of underground excavations. Tunn Undergr Sp Technol 15:117–124. https://doi.org/10.1016/S0886-7798(00)00036-5
Lee JH, Lee BS (2012) Modal analysis of carbon nanotubes and nanocones using FEM. Comput Mater Sci 51:30–42. https://doi.org/10.1016/j.commatsci.2011.06.041
Li H, Liu Y, Liu L, et al (2019a) Numerical evaluation of topographic effects on seismic response of single-faced rock slopes. Bull Eng Geol Environ 78:1873–1891. https://doi.org/10.1007/s10064-017-1200-7
Li L qi, Ju N pan, Zhang S, et al (2019b) Seismic wave propagation characteristic and its effects on the failure of steep jointed anti-dip rock slope. Landslides 16:105–123. https://doi.org/10.1007/s10346-018-1071-4
Li L qi, Ju N pan, Zhang S, Deng X xue (2019c) Shaking table test to assess seismic response differences between steep bedding and toppling rock slopes. Bull Eng Geol Environ 78:519–531. https://doi.org/10.1007/s10064-017-1186-1
Liu H, Xu Q, Li Y (2014) Effect of Lithology and Structure on Seismic Response of Steep Slope in a Shaking Table Test. J Mt Sci 11:371–383
Liu H, Xu Q, Li Y, Fan X (2013) Response of high-strength rock slope to seismic waves in a shaking table test. Bull Seismol Soc Am 103:3012–3025. https://doi.org/10.1785/0120130055
Liu X, Deng Z, Liu Y, et al (2019) Study of cumulative damage and failure mode of horizontal layered rock slope subjected to seismic loads. Rock Soil Mech 40:2507–2516. https://doi.org/10.16285/j.rsm.2018.0675
Mineo S, Pappalardo G, Rapisarda F, et al (2015) Integrated geostructural, seismic and infrared thermography surveys for the study of an unstable rock slope in the Peloritani Chain (NE Sicily). Eng Geol 195:225–235. https://doi.org/10.1016/j.enggeo.2015.06.010
Ning Y, Zhang G, Tang H, et al (2019) Process Analysis of Toppling Failure on Anti-dip Rock Slopes Under Seismic Load in Southwest China. Rock Mech Rock Eng 52:4439–4455. https://doi.org/10.1007/s00603-019-01855-z
Owen LA, Kamp U, Khattak GA, et al (2008) Landslides triggered by the 8 October 2005 Kashmir earthquake. Geomorphology 94:1–9. https://doi.org/10.1016/j.geomorph.2007.04.007
Pelekis P, Batilas A, Pefani E, et al (2017) Surface topography and site stratigraphy effects on the seismic response of a slope in the Achaia-Ilia (Greece) 2008 Mw6.4 earthquake. Soil Dyn Earthq Eng 100:538–554. https://doi.org/10.1016/j.soildyn.2017.05.038
Qi S, Lan H, Dong J (2015) An analytical solution to slip buckling slope failure triggered by earthquake. Eng Geol 194:4–11. https://doi.org/10.1016/j.enggeo.2014.06.004
Song D, Che A, Zhu R, Ge X (2019) Natural Frequency Characteristics of Rock Masses Containing a Complex Geological Structure and Their Effects on the Dynamic Stability of Slopes. Rock Mech Rock Eng 52:4457–4473. https://doi.org/10.1007/s00603-019-01885-7
Song D, Chen Z, Dong L, Zhu W (2021a) Numerical investigation on dynamic response characteristics and deformation mechanism of a bedded rock mass slope subject to earthquake excitation. Appl Sci 11:. https://doi.org/10.3390/app11157068
Song D, Chen Z, Ke Y, Nie W (2020) Seismic response analysis of a bedding rock slope based on time, frequency and time-frequency domains. Elsevier B.V
Song D, Liu X, Huang J, et al (2021b) Characteristics of wave propagation through rock mass slopes with weak structural planes and their impacts on the seismic response characteristics of slopes: a case study in the middle reaches of Jinsha River. Bull Eng Geol Environ 80:1317–1334. https://doi.org/10.1007/s10064-020-02008-1
Song D, Liu X, Li B, et al (2021c) Assessing the influence of a rapid water drawdown on the seismic response characteristics of a reservoir rock slope using time–frequency analysis. Acta Geotech 16:1281–1302. https://doi.org/10.1007/s11440-020-01094-5
USGS (2022) https://www.usgs.gov/. In: United States Geol. Surv.
Wasowski J, Keefer DK, Lee CT (2011) Toward the next generation of research on earthquake-induced landslides: Current issues and future challenges. Eng Geol 122:1–8. https://doi.org/10.1016/j.enggeo.2011.06.001
Xue Q (2000) Need of performance-based earthquake engineering in Taiwan: A lesson from the Chichi earthquake. Earthq Eng Struct Dyn 29:1609–1627. https://doi.org/10.1002/1096-9845(200011)29:11<1609::AID-EQE977>3.0.CO;2-M
Yagoda-Biran G, Hatzor YH (2013) A new failure mode chart for toppling and sliding with consideration of earthquake inertia force. Int J Rock Mech Min Sci 64:122–131. https://doi.org/10.1016/j.ijrmms.2013.08.035
Yang G, Wu F, Dong J, Qi S (2012) Study of Dynamic Response Characters and Failure Mechanism of Rock Slope. Chinese J Rock Mech Eng 31:696–702
Yang Y, Xu D, Zheng H, et al (2021) Modeling Wave Propagation in Rock Masses Using the Contact Potential-Based Three-Dimensional Discontinuous Deformation Analysis Method. Rock Mech Rock Eng 54:2465–2490. https://doi.org/10.1007/s00603-020-02359-x
Yuan R mao, Tang CL, Deng Q hai (2015) Effect of the acceleration component normal to the sliding surface on earthquake-induced landslide triggering. Landslides 12:335–344. https://doi.org/10.1007/s10346-014-0486-9
Zhan Z, Qi S (2017) Numerical Study on dynamic response of a horizontal layered-structure rock slope under a normally incident Sv wave. Appl Sci 7:. https://doi.org/10.3390/app7070716
Zheng Y, Wang R, Chen C, et al (2021) Dynamic analysis of anti-dip bedding rock slopes reinforced by pre-stressed cables using discrete element method. Eng Anal Bound Elem 130:79–93. https://doi.org/10.1016/j.enganabound.2021.05.014

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Numerical investigation on seismic dynamic response characteristics of high-steep layered granite slopes via time-frequency analysis

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Abstract

Figures

1. Introduction

2. Construction Of Three-dimensional Model Slope

2.1 Case study

2.2 Numerical modelling

3. Dynamic Response Characteristics Of Slope Based On Time Domain

3.1 Characteristics of wave propagation in rock mass

3.2 Effect of topography and geology on the dynamic response of slopes

3.3 Influence of input direction of seismic wave

3.4 Dynamic stability analysis of slope

4. Dynamic Response Characteristics Of Slope Based On Frequency Domain

4.1 Modal analysis

4.2 Fourier spectrum analysis

4.3 Dynamic deformation mode analysis of slopes

4.4 Discussion

5 Conclusion

Declarations

References

Additional Declarations

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