Keywords

1 Introduction

The seismic landslides are common seismic hazard phenomena in mountainous areas and have caused many casualties and property losses. Local site conditions have significant effects on ground motion characteristics and earthquake intensity [1,2,3], including topography (slopes, ridges, and canyons) and geology (sedimentation, basins, and faults).

The earthquakes are one of the most common natural disasters that cause ground movement and damage. The slope shapes may have significant topographic and soil effects during earthquakes, exacerbating the damage caused by earthquakes. Therefore, it is crucial to study the influence of slope morphology on topographic and soil effects. There has been a lot of abroad researchers’ attention to this problem, and some results have been achieved. For example, a study by R. Tripe et al. [4] shown that the topographic and soil effects were interactive and should not be treated separately. Zhang et al. [5] shown that the amplification factors at the slope angle of 32.3° were maximum. Ashford et al. [6] proposed that the slope dynamic response can be subdivided into topographic amplification, site amplification, and surface amplification effects. They also derived the corresponding calculations that the topographic amplification factors reach their maximum value at H/λ = 0.2, about 1.5. George et al. [7] shown that the vertical motion components generated within the slopes may be as large as the horizontal. It was due to the reflection of seismic waves on the slope surface.

The domestic scholars have also conducted a lot of dynamic response research on the influence of slope morphology. Through shaking table and numerical analysis methods, Qi Shengwen et al. [8] obtained that the slopes had two types of dynamic response: high slope and low slope typology. They proposed the concept of the critical height of the slopes. The results of the study by Yan Zhixin et al. [9] showed that the rocky slopes’ dynamic responses with different shapes had different deformation damage sites, which was closely related to the slope shapes. Zhang Yingbin et al. [10] analyzed three factors: the slope heights, angles, and shapes. The results showed that different slope shapes caused different acceleration response patterns, and the concave slopes’ acceleration amplification effects were slightly minor. Zhang Yihao et al. [11] analyzed the different slope types’ stabilities and damages of the landslides. The test results shown that the convex slopes required the lowest lifting angles for the same slope gradient and height, and their stabilities were the worst; the concave slopes were the highest, and their stabilities were the best. In addition to this, graded starts were more likely to occur when convex slopes break down.

Although previous studies have yielded some results regarding the dynamic response characteristics of slope under the action of the seismic wave, few studies have been conducted to explore the influence of slope morphology. In this paper, based on the previous work, we used finite difference software to establish three kinds of slope models (straight, concave, and convex slopes) with the same elevation for numerical simulation. And the amplification factors of the slope surface centroid amplification were observed. The laws of the slope dynamic response were summarized when the slope angle, morphology, and seismic wave frequency changed. The mechanism was then analyzed by combining the ray analysis and Snell's law formula derivation.

2 Numerical Simulation

2.1 Soil Parameters

A series of numerical simulations were carried out using the finite difference code FLAC 5.0 [12] to describe the dynamic response characteristics of slope surface centroid amplification under the action of the seismic wave. In the calculation models, the bedrock's upper was a homogeneous, linearly elastic soil layer. A local damping of 0.05 was used to improve the computational efficiency. The soil layer's primary material parameters are shown in Table 1.

Table 1. The physical and mechanical parameters of the soil layer
Full size table

2.2 Model Dimensions and Boundary Conditions

Different slope models are needed to correspond to different working conditions to study the slope dynamic responses under the seismic action. The shear waves (SV waves) come into the slope vertically from the bottom in this study, and the slope models were modeled using the fixed slope height H = 50 m, width 1000 m (\({L}_{1}={L}_{2}\)), and depth of 40m. The 2D models with different slope shapes are shown in Figs. 1, 2 and 3.

Typically, observation points are located at the model free-field boundaries’ left or right, which are used as reference points [4, 13] to calculate the topographic amplification factors. Therefore, this study used the ratio of the peak acceleration at the 2D models’ slope surface center to the 1D models at the same height to calculate the amplification factors. The 1D models were constructed with the same material, depth, load, and boundary conditions as the 2D models. To ensure the computational efficiency and accuracy of the results, the quiet and free-field boundaries (Fig. 1) were used to absorb the approach borders’ waves, preventing any waves from reflecting in the models. The dimensions of the grid (\(\Delta l\)) were chosen according to the recommendations of Kuhlemeyer and Lysmer [14]:

$$ \vartriangle {\textit{l}} \le \frac{\lambda }{10} $$
(1)

where λ is the wavelength corresponding to the highest frequency of the input seismic wave.

Fig. 1.
figure 1

Schematic diagram of the 2D straight slope models

Full size image
Fig. 2.
figure 2

Schematic diagram of the 2D convex slope models

Full size image
Fig. 3.
figure 3

Schematic diagram of the 2D concave slope models

Full size image

2.3 The Input Seismic Wave

The input of the seismic wave was obtained using the Gabor wave [15] with a peak ground acceleration of 0.5 m/\({\text{s}}^{2}\), given by Eq. (2). To mimic the accumulation and attenuation of the seismic waves during an actual seismic event, the modified Gabor wave at the slopes’ bottom was considered. The constant in Eq. (2) was varied so that the acceleration reached 1.0 m/\({\text{s}}^{2}\) (Eq. 3), and the same number of cycles (N = 12) was used for the different frequencies considered. The adjusted seismic acceleration time history profile was shown in Fig. 4.

$$ a(t) = \sqrt {\alpha e^{ - \beta } t^{\gamma } } \sin (2\pi ft) $$
(2)
$$ a(t) = 2\sqrt {\alpha e^{ - \beta } t^{\gamma } } \sin (2\pi ft) $$
(3)

where f is the frequency of the input seismic wave; t is the time; α, β, and γ are parameters regulating the shape and amplitude of the time history envelope of the accelerated seismic wave. Details of the values taken were shown in Table 2.

Fig. 4.
figure 4

Time history diagram of ground shaking acceleration (f = 2Hz)

Full size image
Table 2. Parameters controlling the shape of the acceleration time history
Full size table

2.4 Observation Points and Working Conditions

In order to monitor the input seismic wave accuracy, the frequencies (including the soil intrinsic frequency) were selected to allow the calculation models to produce topographic and soil layer effects. The natural frequencies were calculated by Eq. (4):

$$ f_{n} = \frac{{(2n + 1)V_{s} }}{4Z},n = 0,1,2,...\infty $$
(4)

Observation points were set at the slope models’ bottom, surface center, and crest. And evaluated the slope dynamic response characteristics for the operating conditions listed in Table 3.

3 Numerical Simulation Results and Analysis

The \({a}_{max}\) was the peak acceleration at the 2D slope model surface center, and the \({a}_{h,ff}\) was at the same height in the 1D model.

Topographic amplification factor:

$$ A_{\max } = a_{\max } /a_{h.ff} $$
(5)

Horizontal topographic amplification factor:

$$ A_{h\max } = a_{h\max } /a_{h.ff} $$
(6)

3.1 Verification of Numerical Simulation Results

As shown in Fig. 5, the monitored horizontal topographic amplification factors \({A}_{hmax}\) at the slope crest (the straight slope of Z = 125 m) were compared with the literature [7]. Although the calculation results were not the same (due to the different model sizes, soil parameters, and input seismic wave types), the overall trend was consistent with the previous results (the amplification factors all reach the maximum value of about 1.5 or so when H/λ = 0.2), which verified the input seismic wave reasonableness.

Table 3. Numerical simulation cases
Full size table
Fig. 5.
figure 5

Topographic amplification factors at the slope crest for Z = 125 m and results from Bouckovalas and Papadimitriou.

Full size image

3.2 Effects of the Seismic Wave Frequency

Figure 6 shown the trend of topographic magnification factors with H/λ at the straight slopes’ crest and surface center at different slope angles (45°). When H/λ = 0.2, the amplification factors at the slope crest were the largest, and the maximum value reached about 1.5; when H/λ = 0.5, the amplification factors at the slope surface center were the largest, with the maximum value of about 2.5. This indicates that with different frequencies, the acceleration response of each part of the slope is different and characterized by rhythmic changes along the slope. The amplification factors’ maximum value at the slope surface center was larger than the crest in this study. So, it is necessary to investigate how the slope surface centroid amplification factors change under the action of the seismic wave.

Fig. 6.
figure 6

Amplification factor of ground shaking at the slope crest and the slope center

Full size image

3.3 Effects of the Slope Angle

As shown in Fig. 7, when the straight slopes’ elevation Z = 125 m (H/λ = 0.5), the size relationship between the amplification factors at different slope angles was 30° > 75° > 45°. This result is consistent with the conclusion of the literature [5] that the straight slope topographic amplification factors are maximum at the slope angle of 30°.

Fig. 7.
figure 7

The straight slope surface centroid amplification factors

Full size image

3.4 Effects of the Slope Morphology

Figure 8 was the line graphs of the concave, straight, and convex slopes’ amplification factors (H/λ = 0.5). From Fig. 10, it could be seen that for the slope angle of 75°, the amplification factors were concave > convex > straight slopes; for the slope angle of 45°: convex > straight > concave slopes; for the slope angle of 30°: straight > convex > concave slopes. On the other hand, the size relationship of the concave slope amplification factors at different slope angles was obviously different from the straight slopes (the convex slopes: 30° > 75° > 45°; the concave slopes: 75° > 30° > 45°). These results show that the amplification factors fluctuate as the slope morphology changes.

Fig. 8.
figure 8

The slope surface centroid amplification factors (H/λ = 0.5)

Full size image

4 Discussions

4.1 Theoretical Analysis

Based on the wave propagation, reflection, and diffraction physical phenomena. When an incident SV wave impinges on the free surface, reflected SV and P waves will be generated. This is due to the coupling effects between the P and SV waves caused by the displacement and continuity conditions on the free surface [16]. From Fig. 9, it can be seen that the \({\theta }_{0}\) (=α) is the angle of the SV waves incidence, and the \({\theta }_{1}\) and \({\theta }_{2}\) are the angles of the SV and P waves reflection, respectively. In order to satisfy the boundary displacements and stress conditions on the free surface, the straightforward relation can be derived from the Snell's law as follows [16]:

$$ \sin \theta_{0} = k^{ - 1} \sin \theta_{2} $$
(7)
$$ \theta_{0} = \theta_{1} $$
(8)

where k is a material constant defined by the ratio between the P and SV wave velocity:

$$ k = \frac{{V_{P} }}{{V_{S} }} = \left[ {\frac{2(1 - \nu )}{{1 - 2\nu }}} \right]^{\frac{1}{2}} $$
(9)

where \(\nu \) is the Poisson's ratio, and k = 2 when \(\nu \) = 1/3.

Fig. 9.
figure 9

Propagation of the incident SV, reflected SV and P waves, and the diffracted Rayleigh waves in the slope model [5].

Full size image

In this paper, the numerical analysis was carried out for the working conditions with the slope angle α ≤ 75°. When the SV waves’ reflection angle is \({\theta }_{1}\) ≤ 75°, the SV waves reflected on the inclined plane cannot reach the upper surface; when the reflected P waves’ angle is \({\theta }_{2}\) > 90° - \({\theta }_{0}\), the reflected P waves will reach the upper surface. For \({\theta }_{2}\) = 90° – \({\theta }_{0}\), the oblique angle can be calculated as 26.6° using Eq. (8); for \({\theta }_{2}\) ≥ 90°, the reflected P waves propagate along the inclined plane, and the critical oblique angle is 30° when \({\theta }_{2}\) = 90°. Thus, when 26.6° < α < 30°, the reflected P waves can reach the upper surface directly. It can superimpose with the incoming SV and Rayleigh waves. The amplitude of the reflected P waves is determined by the slope angle, as shown in Eq. (10), and \({A}_{0}\) was the amplitude value of the incident SV waves.

$$ A_{1} = - A_{0} \frac{k\sin 4\alpha }{{\sin 2\theta_{2} \sin 2\alpha + k^{2} \cos^{2} 2\alpha }} $$
(10)

Substituting Eqs. (7) (8) (9) into the above equation yielded that:

$$ \frac{{A_{1} }}{{A_{2} }} = - \frac{k\sin 4\alpha }{{4\sin \alpha \sin 2\alpha \sqrt {1 - 4\sin^{2} \alpha } + k^{2} \cos^{2} 2\alpha }} $$
(11)

From the variation of \(\frac{{A}_{1}}{{A}_{0}}\) with slope angle in Fig. 10, it can be seen that when the slope angle was less than 30°, the amplitude of the reflected P waves increased with the slope angle, and the amplification factors increased as well. As long as the slope angle was kept above 30°, the amplitude of the reflected P waves was close to zero. Therefore, the slope surface centroid amplification factors were reduced somewhat. This is because when the slope angle exceeds 30°, the reflected P waves are transformed into a surface wave propagating along the free surface. It explains why the straight slope amplification factors are maximum at the slope angle of 30°.

Fig. 10.
figure 10

Variation of the amplitude ratio of the reflected P wave with the angle of incidence. (The arrow points to the critical angle of 30°)

Full size image

4.2 Ray Analysis

Under the action of micro-vibration, according to the fluctuation theory of the seismic wave propagation, the rays are used to indicate the seismic waves’ propagation direction, and the sparseness of the wave rays can reflect the convergence or dispersion of the seismic waves propagation energy. From the analysis of Snell's law [16] in Sect. 4.1. A, the distribution of the seismic fluctuation energy in an isotropic uniform slope is determined by the incident seismic wave angle, independent of the vibration direction of the waves. The following will analyze and verify the numerical simulation results based on the seismic wave ray path distribution.

Fig. 11.
figure 11

Schematic diagrams of the straight, concave, and convex slope ray paths

Full size image

It can be seen from Fig. 11(a) that the convex slopes with the angle of 30° had stronger energy in the slopes’ middle and lower regions and weaker energy in the upper part; the straight slopes had more vital energy near the slope surface; and the concave slopes were evanescent to the reflected waves. This was consistent with the numerical simulation results. The amplification factors of the straight slopes were most significant when the slope angle was 30°. It can be seen through Fig. 11(b) that the energy convergence regions caused by convex slopes moved to the slope middle and front. The energy was stronger at the slope surface center. Consistent with the simulation results at the slope angle of 45°, the convex slope amplification factors were the maximum. It can be seen from Fig. 11(c) that when the slope angle was 75°, due to the concave slopes’ dispersion effects on the reflected waves and the influence of the high slope angle, the energy convergence phenomenon in the areas near the slope middle and lower part surface was partially caused. In contrast, the convex slopes’ convergence effects on the reflected waves were mainly reflected behind the slope crest, and there was no apparent convergence phenomenon near the slope surface center. It was also consistent with the above numerical simulation results for the slope angle of 75°, and the concave slope amplification factors were the maximum.

The above results of ray analysis and numerical simulation corroborated each other, indicating that the slope shapes had significant effects on the slope dynamic response. In summary, the influence of slope shapes on the slope dynamic response is not generalized in practical slope problems. It depends on the slopes’ specific conditions.

5 Conclusions

Based on the above numerical simulation and theoretical analysis results, the effects of the slope angle, slope morphology, and seismic wave frequency on the effects on the slope dynamic responses are revealed. The conclusions are drawn as followed.

  1. (1)

    The acceleration responses are different at the various parts of the slope for different frequencies, and the amplification factor shows a rhythmic phenomenon along the elevation.

  2. (2)

    The amplification factors are different for the slope with different slope aspect shape. The slope morphology has a significant effect on the dynamic response of the slope.

  3. (3)

    If the slope elevation is the same and the H/λ equals to 0.5, the amplification factor of straight slope is biggest for the slope angle of 30° and it is smallest for the slope angle of 45°.The slope angle has greatly influences the slope amplification factor.