Dynamic Analysis of the Seismo-Dynamic Response of Anti-Dip Bedding Rock Slopes Using a Three-Dimensional Discrete-Element Method
<p>Side view of the discrete-element model of the slope.</p> "> Figure 2
<p>Three-dimensional view of the slope model.</p> "> Figure 3
<p>Schematic diagrams illustrating the dynamic boundary conditions imposed on the 3D discrete-element models showing: (<b>a</b>) a side view of the model, and (<b>b</b>) a 3D spatial view.</p> "> Figure 4
<p>The input acceleration history of the sinusoidal wave used. The holding time is assumed to be 5 s, the amplitude 3.14 m/s<sup>2</sup>, and the period (<span class="html-italic">T</span>) 0.2 s.</p> "> Figure 5
<p>The nature of the input acceleration used in this work (Wolong waves of the Wenchuan earthquake), showing: (<b>a</b>) the acceleration history with a holding time of 5 s and E–W, U–P, and N–S PGA values of 9.28, 6.39, and −6.08 m/s<sup>2</sup>, respectively, and (<b>b</b>) the corresponding Fourier spectra and earthquake dominant frequency range (1.5–6 Hz).</p> "> Figure 6
<p>Spatial locations of the acceleration (A) and displacement (D) monitoring points and the profiles upon which they are located in the slope model.</p> "> Figure 7
<p>Variation of the acceleration amplification factors at different acceleration monitoring points due to the action of natural seismic and sinusoidal waves for points along profiles: (<b>a</b>) 1, (<b>b</b>) 2, (<b>c</b>) 3, and (<b>d</b>) 4.</p> "> Figure 8
<p>Variation of the peak acceleration amplification factors at four different points in slopes subjected to the action of sinusoidal waves. The values are plotted as functions of the angle <span class="html-italic">γ</span> and the points chosen are: (<b>a</b>) A5 and A9, and (<b>b</b>) A7 and A8.</p> "> Figure 9
<p>Variation of the three-directional peak AAF values at point A8 with slope angle <span class="html-italic">α</span> and joint angle <span class="html-italic">β</span>: (<b>a</b>) under the action of sinusoidal waves, and (<b>b</b>) under the action of natural seismic waves.</p> "> Figure 10
<p>Fourier spectra components and their dominant frequencies at different elevations within a slope subjected to the action of a natural seismic wave. The data are arranged in order of increasing elevation from bottom to top (points A12, A10, and A8 along profile 2).</p> "> Figure 11
<p>Contour plots of the displacement in the <span class="html-italic">Y</span>-direction of a slope (<span class="html-italic">α</span> = 40° and <span class="html-italic">β</span> = 60°) subjected to a natural seismic wave. The plots show the displacements in the plane corresponding to <span class="html-italic">x</span> = 10 m at different times after the onset of the seismic disturbance: (<b>a</b>) 1 s, (<b>b</b>) 2 s, (<b>c</b>) 3 s, (<b>d</b>) 4 s, (<b>e</b>) 5 s, and (<b>f</b>) 6 s.</p> "> Figure 12
<p>Contour plots of the displacement in the <span class="html-italic">Y</span>-direction of a slope (<span class="html-italic">α</span> = 40° and <span class="html-italic">β</span> = 70°) subjected to a sinusoidal wave. The plots show the displacements in the plane corresponding to <span class="html-italic">x</span> = 10 m at the end of the period of disturbance (<span class="html-italic">t</span> = 5 s) for different values of <span class="html-italic">γ</span>: (<b>a</b>) 0°, (<b>b</b>) 5°, (<b>c</b>) 10°, (<b>d</b>) 15°, and (<b>e</b>) 20°.</p> "> Figure 13
<p>Example of the use of Newmark’s method: (<b>a</b>) acceleration history (<span class="html-italic">Y</span>-direction) at point A6, (<b>b</b>) corresponding relative velocity history, and (<b>c</b>) cumulative displacement history.</p> "> Figure 14
<p>Displacement history of point D5 in the <span class="html-italic">Y</span>-direction for different slopes subjected to the action of a natural seismic wave. The slopes are labeled <span class="html-italic">α</span>_<span class="html-italic">β</span> according to their <span class="html-italic">α</span> and <span class="html-italic">β</span> angles.</p> "> Figure 15
<p>Displacement history of point D5 in the <span class="html-italic">Y</span>-direction for slopes subjected to the action of a sinusoidal wave. Each slope has <span class="html-italic">α</span> = 40° and <span class="html-italic">β</span> = 70° but different values of <span class="html-italic">γ</span>.</p> "> Figure 16
<p>Comparison of the permanent displacements of point D5 (in the <span class="html-italic">Y</span>-direction) under the action of a natural seismic wave as calculated using Newmark’s method and dynamic 3D discrete-element method.</p> ">
Abstract
:1. Introduction
2. Numerical Modeling Using 3D Discrete Elements
2.1. Overview
2.2. Setting the Dynamic Parameters
3. Dynamic Response of the Slope
3.1. Analysis of the Amplification Effect
3.2. Fourier Spectra
4. Mechanism Responsible for Slope Failure under Seismic Action
5. Permanent Displacement of the Slope
5.1. Newmark’s Method
5.2. Displacements Calculated Using the 3D Discrete-Element Method
5.3. Comparison of Results and Discussion
- Compared with bedding slopes (or landslides), anti-dip bedding slopes are more stable under static conditions and have higher safety factors (SF). On the other hand, Newmark’s method is based on the assumption that sliding failure occurs with a straight failure surface. This is obviously contrary to the flexure toppling failure mode of anti-dip bedding rock slopes.
- Newmark’s method is based on a pseudo-static method and assumes that the geotechnical object can be regarded as a rigid, nondeforming body. In reality, the geotechnical object will deform and the 3D discrete-element method takes this into account. As a result, the displacement obtained using our method will be larger.
- Newmark’s method is based on the assumption that the problem can be treated in a single plane; the seismic waves and displacements are also considered to be in the same horizontal direction. In this paper, we consider the loading caused by the seismic waves in the X-, Y- and Z-directions. This means our model can account for 3D spatial effects that cannot be treated using a simple single-plane approach.
6. Conclusions
- When the slope is dynamically loaded using a 3D seismic wave, the 3D acceleration magnification coefficient of the slope is not very regular; there is also no sign of obvious ‘elevation’ and ‘appearance’ effects. Instead, it shows rhythmicity. The peak acceleration amplification effect is more significant under the action of sinusoidal waves. As the angle between the joint and slope trends increases, the amplification effect of the slope first decreases and then increases.
- As the seismic wave propagates within the slope, the amplitudes of the low-frequency parts of the seismic wave are generally amplified and the earthquake dominant frequency tends to decrease. The EDF in the Y-direction also diverges, forming 2 or 3 peaks.
- When subjected to seismic action, the slope starts to fail in the lower part of the slope and failure progresses from bottom to top. The failure surface is step-like, showing the characteristics of flexure toppling. When the angle between the joint and slope trends is nonzero, the failure surface becomes more irregular. Moreover, as this angle increases, the depth of the failure surface increases and the scope of the unstable part of the slope becomes wider.
- The permanent displacement of the slope increases as the slope and joint angles increase. Under the action of natural seismic waves, the displacement of the slope in the Y-direction develops slowly at first and then accelerates at a critical time Tc. As the angle between the joint and slope trends increases, the permanent displacement of the slope increases and then decreases.
- Newmark’s method only considers the action of seismic waves in one direction (i.e., the method is only useful for planar problems). The displacement results obtained using the method are much smaller than those obtained using the dynamic 3D discrete-element method. The results obtained using Newmark’s method are therefore overly conservative.
Author Contributions
Funding
Conflicts of Interest
References
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Density (kN/m3) | Young’s Modulus (GPa) | Poisson’s Ratio | Cohesion (MPa) | Friction Angle (°) | Tensile Strength (MPa) | Dilation Angle (°) | Normal Stiffness (GPa/m) | Tangential Stiffness (GPa/m) | |
---|---|---|---|---|---|---|---|---|---|
Rock mass | 25.97 | 10.5 | 0.25 | 1.5 | 45 | 0.5 | 0 | – | – |
Structural planes | – | – | – | 0.24 | 32 | 0.001 | 0 | 58.33 | 19.44 |
Model Number | α (°) | β (°) | SF | N (m/s2) | D (cm) |
---|---|---|---|---|---|
1 | 40 | 50 | 7.88 | 5.77 | 0.00030 |
2 | 40 | 60 | 6.46 | 4.58 | 0.00090 |
3 | 40 | 70 | 3.99 | 2.51 | 0.12 |
4 | 50 | 50 | 5.66 | 5.55 | 0.043 |
5 | 50 | 60 | 4.07 | 3.66 | 0.015 |
6 | 50 | 70 | 2.71 | 2.04 | 0.76 |
7 | 60 | 50 | 3.95 | 5.11 | 0.72 |
8 | 60 | 60 | 2.70 | 2.94 | 0.15 |
9 | 60 | 70 | 2.02 | 1.77 | 0.64 |
1 | 40 | 50 | 7.88 | 5.77 | 0.00030 |
2 | 40 | 60 | 6.46 | 4.58 | 0.00090 |
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Ren, Z.; Chen, C.; Sun, C.; Wang, Y. Dynamic Analysis of the Seismo-Dynamic Response of Anti-Dip Bedding Rock Slopes Using a Three-Dimensional Discrete-Element Method. Appl. Sci. 2022, 12, 4640. https://doi.org/10.3390/app12094640
Ren Z, Chen C, Sun C, Wang Y. Dynamic Analysis of the Seismo-Dynamic Response of Anti-Dip Bedding Rock Slopes Using a Three-Dimensional Discrete-Element Method. Applied Sciences. 2022; 12(9):4640. https://doi.org/10.3390/app12094640
Chicago/Turabian StyleRen, Zhanghao, Congxin Chen, Chaoyi Sun, and Yue Wang. 2022. "Dynamic Analysis of the Seismo-Dynamic Response of Anti-Dip Bedding Rock Slopes Using a Three-Dimensional Discrete-Element Method" Applied Sciences 12, no. 9: 4640. https://doi.org/10.3390/app12094640